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Deck 7: Functions of Several Variables
Full screen (f)
Question
Sketch the graph of the plane given by 
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Sketch the graph of the plane given by 
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
The two planes 
and 
are perpendicular?
and are perpendicular? Question
Sketch the graph of the plane given by 
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Sketch the yz-trace of the equation: 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Sketch the trace of the intersection of plane 
with the sphere: 
A) 
B) 
C) 
D) 
E) 
with the sphere: A)
B)
C)
D)
E)
Question
Find the center and radius of the sphere. 
A) Center:
Radius:
B) Center:
Radius:
C) Center:
Radius:
D) Center:
Radius:
E) Center:
Radius:
A) Center:
Radius:
B) Center:
Radius:
C) Center:
Radius:
D) Center:
Radius:
E) Center:
Radius:
Question
Sketch the yz-trace of the equation: 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the equation of the sphere that has the points 
and 
as endpoints of a diameter.
A)
B)
C)
D)
E)
and as endpoints of a diameter. A)
B)
C)
D)
E)
Question
Sketch the trace of the intersection of plane 
with the sphere: 
A) 
B) 
C) 
D)
E) 
with the sphere: A)
B)
C)
D)
E)
Question
Find the intercepts of the plane given by 
A) The x-intercept is
The y-intercept is 
The z-intercept is 
B) The x-intercept is
The y-intercept is 
The z-intercept is 
C) The x-intercept is
The y-intercept is 
The z-intercept is 
D) The x-intercept is
The y-intercept is 
The z-intercept is
E) The x-intercept is
The y-intercept is 
The z-intercept is 
A) The x-intercept is
The y-intercept is The z-intercept is B) The x-intercept is
The y-intercept is The z-intercept is C) The x-intercept is
The y-intercept is The z-intercept is D) The x-intercept is
The y-intercept is The z-intercept is
E) The x-intercept is
The y-intercept is The z-intercept is Question
Find 
if the midpoint of the line segment joining the two points 
and 
is 
A)
B)
C)
D)
E)
if the midpoint of the line segment joining the two points and is A)
B)
C)
D)
E)
Question
Find the standard equation of the sphere whose center is 
and whose radius is 5.
A)
B)
C)
D)
E)
and whose radius is 5. A)
B)
C)
D)
E)
Question
Sketch the yz-trace of the equation: 
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the center and radius of the sphere whose equation is 
Round your answer to two decimal places,where applicable.
A) center:
; radius: 2.00
B) center:
; radius: 2.00
C) center:
; radius: 8.00
D) center:
; radius: 4.00
E) center:
; radius: 8.00
Round your answer to two decimal places,where applicable. A) center:
; radius: 2.00B) center:
; radius: 2.00C) center:
; radius: 8.00D) center:
; radius: 4.00E) center:
; radius: 8.00 Question
Find the lengths of the sides of the triangle with the given vertices,and determine whether the triangle is a right triangle,an isosceles triangle,or neither. 
A)
; obtuse triangle
B)
; right triangle
C)
; right triangle
D)
; acute triangle
E)
; acute triangle
A)
; obtuse triangleB)
; right triangleC)
; right triangleD)
; acute triangleE)
; acute triangle Question
Find the intercepts of the plane given by 
A) The x-intercept is
The z-intercept is 
B) The x-intercept is
The z-intercept is 
C) The x-intercept is
The z-intercept is 
D) The x-intercept is
The z-intercept is 
E) The x-intercept is
The z-intercept is 
A) The x-intercept is
The z-intercept is B) The x-intercept is
The z-intercept is C) The x-intercept is
The z-intercept is D) The x-intercept is
The z-intercept is E) The x-intercept is
The z-intercept is Question
Sketch the trace of the intersection of plane 
with the sphere: 
A) 
B) 
C) 
D)
E) 
with the sphere: A)
B)
C)
D)
E)
Question
Find the the distance between the two points 
and 
A) 1 unit
B) 6 units
C)
units
D) 8 units
E) 38 units
and A) 1 unit
B) 6 units
C)
unitsD) 8 units
E) 38 units
Question
Because of the forces caused by its rotation,a planet is actually an oblate ellipsoid rather than a sphere.The equatorial radius is 3967 miles and the polar radius is 3951 miles.Find an equation of the ellipsoid.Assume that the center of a planet is at the origin and the xy- trace 
corresponds to the equator.
A)
B)
C)
D)
E)
corresponds to the equator. A)
B)
C)
D)
E)
Question
Identify the quadric surface. 
A) The graph is hyperboloid of two sheets.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is an elliptic paraboloid.
E) The graph is a hyperboloid of one sheet.
A) The graph is hyperboloid of two sheets.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is an elliptic paraboloid.
E) The graph is a hyperboloid of one sheet.
Question
Sketch the level curves for the function below for the given c-values 
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the domain and range of the function. 
A) Domain: all point
such that 
Range:
B) Domain: all point
such that 
Range:
C) Domain: all point
such that 
Range:
D) Domain: all point
such that 
Range:
E) Domain: all point
such that 
Range:
A) Domain: all point
such that Range:
B) Domain: all point
such that Range:
C) Domain: all point
such that Range:
D) Domain: all point
such that Range:
E) Domain: all point
such that Range:
Question
Describe the trace of the surface given by the function below 
in the xz-plane.
A) circle
B) parabola
C) line
D) ellipse (not circle)
E) hyperbola
in the xz-plane. A) circle
B) parabola
C) line
D) ellipse (not circle)
E) hyperbola
Question
Evaluate the function at the given values of the independent variables. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
If 
,find 
and 
A)
B)
C)
D)
E)
,find and A)
B)
C)
D)
E)
Question
Describe the trace of the surface given by the function below in the xy-plane. 
x2 + y2 + z2 = 1
A) parabola
B) ellipse (not circle)
C) circle
D) hyperbola
E) line
x2 + y2 + z2 = 1
A) parabola
B) ellipse (not circle)
C) circle
D) hyperbola
E) line
Question
Sketch the level curves for the function below for the given c-values 
A) 
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Describe the domain and range of the function. ![<strong>Describe the domain and range of the function. </strong> A) domain: The disk Range: The interval (0, 1) B) domain: The disk Range: The interval [0, 1] C) domain: The disk Range: The interval [0, 1) D) domain: The disk Range: The interval [0, 1] E) domain: The disk Range: The interval [0,1) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a84_ec77_861b_efb52ce014e9_TB1301_11.jpg)
A) domain: The disk![<strong>Describe the domain and range of the function. </strong> A) domain: The disk Range: The interval (0, 1) B) domain: The disk Range: The interval [0, 1] C) domain: The disk Range: The interval [0, 1) D) domain: The disk Range: The interval [0, 1] E) domain: The disk Range: The interval [0,1) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a85_1388_861b_07bf28e9ac33_TB1301_11.jpg)
Range: The interval (0, 1)
B) domain: The disk![<strong>Describe the domain and range of the function. </strong> A) domain: The disk Range: The interval (0, 1) B) domain: The disk Range: The interval [0, 1] C) domain: The disk Range: The interval [0, 1) D) domain: The disk Range: The interval [0, 1] E) domain: The disk Range: The interval [0,1) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a85_1389_861b_993ca16bfc11_TB1301_11.jpg)
Range: The interval [0, 1]
C) domain: The disk![<strong>Describe the domain and range of the function. </strong> A) domain: The disk Range: The interval (0, 1) B) domain: The disk Range: The interval [0, 1] C) domain: The disk Range: The interval [0, 1) D) domain: The disk Range: The interval [0, 1] E) domain: The disk Range: The interval [0,1) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a85_138a_861b_29840b871946_TB1301_11.jpg)
Range: The interval [0, 1)
D) domain: The disk![<strong>Describe the domain and range of the function. </strong> A) domain: The disk Range: The interval (0, 1) B) domain: The disk Range: The interval [0, 1] C) domain: The disk Range: The interval [0, 1) D) domain: The disk Range: The interval [0, 1] E) domain: The disk Range: The interval [0,1) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a85_138b_861b_e354c76f8ad5_TB1301_11.jpg)
Range: The interval [0, 1]
E) domain: The disk![<strong>Describe the domain and range of the function. </strong> A) domain: The disk Range: The interval (0, 1) B) domain: The disk Range: The interval [0, 1] C) domain: The disk Range: The interval [0, 1) D) domain: The disk Range: The interval [0, 1] E) domain: The disk Range: The interval [0,1) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a85_3a9c_861b_bf4c9eef70b7_TB1301_11.jpg)
Range: The interval [0,1)
A) domain: The disk
Range: The interval (0, 1)
B) domain: The disk
Range: The interval [0, 1]
C) domain: The disk
Range: The interval [0, 1)
D) domain: The disk
Range: The interval [0, 1]
E) domain: The disk
Range: The interval [0,1)
Question
Identify the quadric surface. z2 - x2 - 
y2 = 1
A) The graph is a hyperboloid of one sheet.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is hyperboloid of two sheets.
E) The graph is an elliptic paraboloid.
y2 = 1A) The graph is a hyperboloid of one sheet.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is hyperboloid of two sheets.
E) The graph is an elliptic paraboloid.
Question
The two planes 
and 
are parallel?
and are parallel? Question
If 
,find 
and 
A)
, 
B)
, 
C)
, 
D)
, 
E)
, 
,find and A)
, B)
, C)
, D)
, E)
, Question
Use the function 
to find 
A)
B)
C)
D)
E)
to find A)
B)
C)
D)
E)
Question
Describe the level curves for the function 
for the c-values given by 
A)
; 
; 
; 
; 
; 
B) 
; 
; 
; 
; 
; 
C) 
; 
; 
; 
; 
; 
D) 
; 
; 
; 
; 
; 
E) 
; 
; 
; 
; 
; 
for the c-values given by A)
;
;
;
;
; B)
;
;
;
;
; C)
;
;
;
;
; D)
;
;
;
;
; E)
;
;
;
;
; Question
Evaluate the function 
at 
A)
B)
C)
D)
E)
at A)
B)
C)
D)
E)
Question
Sketch the level curves for the function below for the given c-values 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use the function 
to find 
A)
B)
C)
D)
E)
to find A)
B)
C)
D)
E)
Question
A manufacturer estimates the Cobb-Douglas production function to be given by 
Estimate the production levels when
and 
A) 135,540 units
B) 122,560 units
C) 131,601 units
D) 145,330 units
E) 112,745 units
Estimate the production levels when
and A) 135,540 units
B) 122,560 units
C) 131,601 units
D) 145,330 units
E) 112,745 units
Question
Describe the level curves of the function.Sketch the level curves for the given c-values. 
, 
A) 
B) 
C) 
D) 
E) 
, A)
B)
C)
D)
E)
Question
The value 
of an investment of $18,000 after t years in an account for which the interest rate r is compounded continuously is given by the function 
dollars.Write the partial derivative 
A)
B)
C)
D)
E)
of an investment of $18,000 after t years in an account for which the interest rate r is compounded continuously is given by the function dollars.Write the partial derivative A)
B)
C)
D)
E)
Question
Find the first partial derivatives with respect to x,y,and z. 
A)
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Evaluate 
and 
for the function 
at the point 
A)
and 
B)
and 
C)
and 
D)
and 
E)
and 
and for the function at the point A)
and B)
and C)
and D)
and E)
and Question
Find the four second partial derivatives.Observe that the second mixed partials are equal. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Test for relative extrema and saddle points. 
A) saddle point at
B) saddle point at
C) saddle point at
D) relative minimum at
E) relative minimum at
A) saddle point at
B) saddle point at
C) saddle point at
D) relative minimum at
E) relative minimum at
Question
Examine the function given below for relative extrema and saddle points. 
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
Question
Examine the function given below for relative extrema and saddle points. 
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a saddle point at
E) The function has a relative maximum at
Question
Examine the function 
for relative extrema.
A) saddle point at
B) relative minimum at
C) saddle point at
D) relative maximum at
E) saddle point at
for relative extrema. A) saddle point at
B) relative minimum at
C) saddle point at
D) relative maximum at
E) saddle point at
Question
Examine the function given below for relative extrema and saddle points. 
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
Question
Examine the function given below for relative extrema and saddle points. 
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
Question
Examine the function 
for relative extrema.
A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
for relative extrema. A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
Question
Evaluate 
and 
for the function 
at the point 
Round your answer to two decimal places.
A)
and 
B)
and 
C)
and 
D)
and 
E)
and 
and for the function at the point Round your answer to two decimal places. A)
and B)
and C)
and D)
and E)
and Question
For 
,find all values of x and y such that 
and 
simultaneously. 
A)
B) 
C) (0,0)
D) Both A and C
E) Both A and B
,find all values of x and y such that and simultaneously. A)
B)
C) (0,0)
D) Both A and C
E) Both A and B
Question
The utility function 
is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is 
Determine the marginal utility of product x.
A)
B) 
C) 
D) 
E) 
is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is Determine the marginal utility of product x. A)
B)
C)
D)
E)
Question
Examine the function given below for relative extrema and saddle points. 
A) The function has a relative maximum at
B) The function has a relative minimum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
A) The function has a relative maximum at
B) The function has a relative minimum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
Question
Find the slopes of the surface 
in the x- and y- directions at the point 
A)
Slope in x-direction: 1
Slope in y-direction: 5
B)
Slope in x-direction: -2
Slope in y-direction: 6
C)
Slope in x-direction: 5
Slope in y-direction: 1
D)
Slope in x-direction: 4
Slope in y-direction: 4
E)
Slope in x-direction: 6
Slope in y-direction: -2
in the x- and y- directions at the point A)
Slope in x-direction: 1
Slope in y-direction: 5
B)
Slope in x-direction: -2
Slope in y-direction: 6
C)
Slope in x-direction: 5
Slope in y-direction: 1
D)
Slope in x-direction: 4
Slope in y-direction: 4
E)
Slope in x-direction: 6
Slope in y-direction: -2
Question
Examine the function 
for relative extrema and saddle points.
A) saddle point:
; relative minimum: 
B) relative minimum:
; relative maximum: 
C) saddle points:
, 
D) saddle point:
; relative minimum: 
E) relative minimum:
; relative maximum: 
for relative extrema and saddle points. A) saddle point:
; relative minimum: B) relative minimum:
; relative maximum: C) saddle points:
, D) saddle point:
; relative minimum: E) relative minimum:
; relative maximum: Question
Examine the function given below for relative extrema and saddle points. 
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
Question
Determine whether there is a relative maximum,a relative minimum,a saddle point,or insufficient information to determine the nature of the function 
at the critical point 
A) insufficient information to determine the nature of the function at
B) relative maximum at 
C) saddle point at
D) relative minimum at
at the critical point A) insufficient information to determine the nature of the function at
B) relative maximum at
C) saddle point at
D) relative minimum at
Question
A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model.The cost function for producing x freestanding and y fireplace-insert stoves is 
Find the marginal costs ( 
and 
)when 
and 
Round your answers to two decimal places.
A)
B) 
C) 
D) 
E) 
Find the marginal costs ( and )when and Round your answers to two decimal places. A)
B)
C)
D)
E)
Question
Find three positive numbers x,y,and z whose sum is 27 and product is a maximum.
A)
B) 
C)
D) 
E) 
A)
B)
C)
D)
E)
Question
Use Lagrange multipliers to maximize the function 
subject to the following constraint: 
Assume that x,y,and z are positive.
A) 110
B)
C)
D) 10
E) no absolute maximum
subject to the following constraint: Assume that x,y,and z are positive.
A) 110
B)
C)
D) 10
E) no absolute maximum
Question
A rectangular box is resting on the xy-plane with one vertex at the origin.The opposite vertex lies in the plane 
Find the dimensions that maximize the volume. (Hint: Maximize 
subject to the constraint 
).
A) 15 units × 12 units × 5 units
B) 11 units × 9 units × 5 units
C) 10 units × 9 units × 6 units
D) 15 units × 10 units × 6 units
E) 12 units × 11 units × 7 units
Find the dimensions that maximize the volume. (Hint: Maximize subject to the constraint ).
A) 15 units × 12 units × 5 units
B) 11 units × 9 units × 5 units
C) 10 units × 9 units × 6 units
D) 15 units × 10 units × 6 units
E) 12 units × 11 units × 7 units
Question
Find the critical points of the function 
,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point.
A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point. A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
Question
Use Lagrange multipliers to find the given extremum.In each case,assume that x and y are positive.
Maximize
Constraint 
A)
B) 
C) 
D) 
E) 
Maximize
Constraint A)
B)
C)
D)
E)
Question
A company manufactures two types of sneakers: running shoes and basketball shoes.The total revenue from x1 units of running shoes and x2 units of basketball shoes is: 
,
Where x1 and x2 are in thousands of units.Find x1 and x2 so as to maximize the revenue.
A)
B)
C)
D)
E)
,
Where x1 and x2 are in thousands of units.Find x1 and x2 so as to maximize the revenue.
A)
B)
C)
D)
E)
Question
Find the critical points of the function 
,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point.
A) relative minima at
, 
, 
, where a, b, c, d, m, and n are arbitrary real numbers
B) relative maxima at
, 
, 
, where a, b, c, d, m, and n are arbitrary real numbers
C) relative minimum at
D) relative maximum at
E) no relative extrema
,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point. A) relative minima at
, , , where a, b, c, d, m, and n are arbitrary real numbersB) relative maxima at
, , , where a, b, c, d, m, and n are arbitrary real numbersC) relative minimum at
D) relative maximum at
E) no relative extrema
Question
An agronomist used four test plots to determine the relationship between the wheat yield y (in bushels per acre)and the amount of fertilizer x (in pounds per acre).The results are shown in the table.
(I)Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data.
(II)Estimate the yield for a fertilizer application of 160 pounds per acre.
A) (I)
; (II) 39.10
B) (I)
; (II) 45.90
C) (I)
; (II) 44.18
D) (I)
; (II) 42.32
E) (I)
; (II)54.94
(I)Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data.
(II)Estimate the yield for a fertilizer application of 160 pounds per acre.
A) (I)
; (II) 39.10B) (I)
; (II) 45.90C) (I)
; (II) 44.18D) (I)
; (II) 42.32E) (I)
; (II)54.94 Question
Find the least squares regression line for the given points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. 
A)
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
A manufacturer has an order for 1,100 units of fine paper that can be produced at two locations.Let x1 and x2 be the numbers of units produced at the two plants.Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by 
A)
units and 
units
B)
units and 
units
C)
units and 
units
D)
units and 
units
E)
units and 
units
A)
units and unitsB)
units and unitsC)
units and unitsD)
units and unitsE)
units and units Question
Use Lagrange multipliers to find the given extremum.In each case,assume that x,y,and z are positive.
Maximize
Constraints 
A)
B)
C)
D)
E)
Maximize
Constraints A)
B)
C)
D)
E)
Question
The sum of the length (denote by z)and the girth (perimeter of a cross section)of packages carried by a delivery service cannot exceed 36 inches.Find the dimensions of the rectangular package of largest volume that may be sent.
A)
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the sum of the squared errors for the linear model 
and the quadratic model 
using the given points. 
A)
B) 
C) 
D) 
E) 
and the quadratic model using the given points. A)
B)
C)
D)
E)
Question
Use Lagrange multipliers to find the given extremum.Assume that x and y are positive.
Maximize
Constraint 
A)
B) 
C) 
D) 
E) 
Maximize
Constraint A)
B)
C)
D)
E)
Question
Use Lagrange multipliers to minimize the function 
subject to the following constraint. 
Assume that x,y,and z are positive.
A) 4
B) 12
C) 8
D) 24
E) 36
subject to the following constraint. Assume that x,y,and z are positive.
A) 4
B) 12
C) 8
D) 24
E) 36
Question
A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals.Fencing for the perimeter costs 10 per foot.To separate the corrals,a fence that costs 8 per foot will divide the region.The total area of the two corrals is to be 7,000 square feet.Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing. 
A) dimensions: 50 feet by 70 feet
B) dimensions: 140 feet by 50 feet
C) dimensions: 25 feet by 70 feet
D) dimensions: 35 feet by 50 feet
E) dimensions: 25 feet by 35 feet
A) dimensions: 50 feet by 70 feet
B) dimensions: 140 feet by 50 feet
C) dimensions: 25 feet by 70 feet
D) dimensions: 35 feet by 50 feet
E) dimensions: 25 feet by 35 feet
Question
A microbiologist must prepare a culture medium in which to grow a certain type of bacteria.The percent of salt contained in this medium is given by 
Where x,y,and z are the nutrient solutions to be mixed in the medium.For the bacteria to grow,the medium must be 13% salt.Nutrient solutions x,y,and z cost $1,$2,and $3 per liter,respectively.How much of each nutrient solution should be used to minimize the cost of the culture medium?
A) 
B) 
C) 
D) 
E) 
Where x,y,and z are the nutrient solutions to be mixed in the medium.For the bacteria to grow,the medium must be 13% salt.Nutrient solutions x,y,and z cost $1,$2,and $3 per liter,respectively.How much of each nutrient solution should be used to minimize the cost of the culture medium?
A)
B)
C)
D)
E)
Question
Use Lagrange multipliers to find the minimum distance from the circle 
to the point 
Round your answer to the nearest tenth.
A) 13.2
B) 160.0
C) 173.7
D) 10.8
E) 116.8
to the point Round your answer to the nearest tenth. A) 13.2
B) 160.0
C) 173.7
D) 10.8
E) 116.8
Question
Find three positive numbers x,y,and z whose sum is 18 and the sum of the squares is a minimum.
A)
B) 
C)
D) 
E) 
A)
B)
C)
D)
E)
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Deck 7: Functions of Several Variables
1
Sketch the graph of the plane given by
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
2
Sketch the graph of the plane given by
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
3
The two planes and are perpendicular?
and are perpendicular? True
4
Sketch the graph of the plane given by
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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5
Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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6
Sketch the yz-trace of the equation:
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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7
Sketch the trace of the intersection of plane with the sphere:
A)
B)
C)
D)
E)
with the sphere: A)
B)
C)
D)
E)
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8
Find the center and radius of the sphere.
A) Center:
Radius:
B) Center:
Radius:
C) Center:
Radius:
D) Center:
Radius:
E) Center:
Radius:
A) Center:
Radius:
B) Center:
Radius:
C) Center:
Radius:
D) Center:
Radius:
E) Center:
Radius:
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9
Sketch the yz-trace of the equation:
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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10
Find the equation of the sphere that has the points and as endpoints of a diameter.
A)
B)
C)
D)
E)
and as endpoints of a diameter. A)
B)
C)
D)
E)
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11
Sketch the trace of the intersection of plane with the sphere:
A)
B)
C)
D)
E)
with the sphere: A)
B)
C)
D)
E)
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12
Find the intercepts of the plane given by
A) The x-intercept is The y-intercept is The z-intercept is
B) The x-intercept is The y-intercept is The z-intercept is
C) The x-intercept is The y-intercept is The z-intercept is
D) The x-intercept is The y-intercept is
The z-intercept is
E) The x-intercept is The y-intercept is The z-intercept is
A) The x-intercept is
The y-intercept is The z-intercept is B) The x-intercept is
The y-intercept is The z-intercept is C) The x-intercept is
The y-intercept is The z-intercept is D) The x-intercept is
The y-intercept is The z-intercept is
E) The x-intercept is
The y-intercept is The z-intercept is Unlock Deck
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13
Find if the midpoint of the line segment joining the two points and is
A)
B)
C)
D)
E)
if the midpoint of the line segment joining the two points and is A)
B)
C)
D)
E)
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14
Find the standard equation of the sphere whose center is and whose radius is 5.
A)
B)
C)
D)
E)
and whose radius is 5. A)
B)
C)
D)
E)
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15
Sketch the yz-trace of the equation:
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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16
Find the center and radius of the sphere whose equation is Round your answer to two decimal places,where applicable.
A) center: ; radius: 2.00
B) center: ; radius: 2.00
C) center: ; radius: 8.00
D) center: ; radius: 4.00
E) center: ; radius: 8.00
Round your answer to two decimal places,where applicable. A) center:
; radius: 2.00B) center:
; radius: 2.00C) center:
; radius: 8.00D) center:
; radius: 4.00E) center:
; radius: 8.00 Unlock Deck
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17
Find the lengths of the sides of the triangle with the given vertices,and determine whether the triangle is a right triangle,an isosceles triangle,or neither.
A) ; obtuse triangle
B) ; right triangle
C) ; right triangle
D) ; acute triangle
E) ; acute triangle
A)
; obtuse triangleB)
; right triangleC)
; right triangleD)
; acute triangleE)
; acute triangle Unlock Deck
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18
Find the intercepts of the plane given by
A) The x-intercept is The z-intercept is
B) The x-intercept is The z-intercept is
C) The x-intercept is The z-intercept is
D) The x-intercept is The z-intercept is
E) The x-intercept is The z-intercept is
A) The x-intercept is
The z-intercept is B) The x-intercept is
The z-intercept is C) The x-intercept is
The z-intercept is D) The x-intercept is
The z-intercept is E) The x-intercept is
The z-intercept is Unlock Deck
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19
Sketch the trace of the intersection of plane with the sphere:
A)
B)
C)
D)
E)
with the sphere: A)
B)
C)
D)
E)
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20
Find the the distance between the two points and
A) 1 unit
B) 6 units
C) units
D) 8 units
E) 38 units
and A) 1 unit
B) 6 units
C)
unitsD) 8 units
E) 38 units
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21
Because of the forces caused by its rotation,a planet is actually an oblate ellipsoid rather than a sphere.The equatorial radius is 3967 miles and the polar radius is 3951 miles.Find an equation of the ellipsoid.Assume that the center of a planet is at the origin and the xy- trace corresponds to the equator.
A)
B)
C)
D)
E)
corresponds to the equator. A)
B)
C)
D)
E)
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22
Identify the quadric surface.
A) The graph is hyperboloid of two sheets.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is an elliptic paraboloid.
E) The graph is a hyperboloid of one sheet.
A) The graph is hyperboloid of two sheets.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is an elliptic paraboloid.
E) The graph is a hyperboloid of one sheet.
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23
Sketch the level curves for the function below for the given c-values
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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24
Find the domain and range of the function.
A) Domain: all point such that
Range:
B) Domain: all point such that
Range:
C) Domain: all point such that
Range:
D) Domain: all point such that
Range:
E) Domain: all point such that
Range:
A) Domain: all point
such that Range:
B) Domain: all point
such that Range:
C) Domain: all point
such that Range:
D) Domain: all point
such that Range:
E) Domain: all point
such that Range:
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25
Describe the trace of the surface given by the function below in the xz-plane.
A) circle
B) parabola
C) line
D) ellipse (not circle)
E) hyperbola
in the xz-plane. A) circle
B) parabola
C) line
D) ellipse (not circle)
E) hyperbola
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26
Evaluate the function at the given values of the independent variables.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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27
If ,find and
A)
B)
C)
D)
E)
,find and A)
B)
C)
D)
E)
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28
Describe the trace of the surface given by the function below in the xy-plane. x2 + y2 + z2 = 1
A) parabola
B) ellipse (not circle)
C) circle
D) hyperbola
E) line
x2 + y2 + z2 = 1
A) parabola
B) ellipse (not circle)
C) circle
D) hyperbola
E) line
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29
Sketch the level curves for the function below for the given c-values
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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30
Describe the domain and range of the function.
A) domain: The disk
Range: The interval (0, 1)
B) domain: The disk
Range: The interval [0, 1]
C) domain: The disk
Range: The interval [0, 1)
D) domain: The disk
Range: The interval [0, 1]
E) domain: The disk
Range: The interval [0,1)
A) domain: The disk
Range: The interval (0, 1)
B) domain: The disk
Range: The interval [0, 1]
C) domain: The disk
Range: The interval [0, 1)
D) domain: The disk
Range: The interval [0, 1]
E) domain: The disk
Range: The interval [0,1)
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31
Identify the quadric surface. z2 - x2 - y2 = 1
A) The graph is a hyperboloid of one sheet.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is hyperboloid of two sheets.
E) The graph is an elliptic paraboloid.
y2 = 1A) The graph is a hyperboloid of one sheet.
B) The graph is an ellipsoid.
C) The graph is an elliptic cone.
D) The graph is hyperboloid of two sheets.
E) The graph is an elliptic paraboloid.
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32
The two planes and are parallel?
and are parallel? Unlock Deck
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33
If ,find and
A) ,
B) ,
C) ,
D) ,
E) ,
,find and A)
, B)
, C)
, D)
, E)
, Unlock Deck
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34
Use the function to find
A)
B)
C)
D)
E)
to find A)
B)
C)
D)
E)
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35
Describe the level curves for the function for the c-values given by
A)
;
;
;
;
;
B) ;
;
;
;
;
C) ;
;
;
;
;
D) ;
;
;
;
;
E) ;
;
;
;
;
for the c-values given by A)
;
;
;
;
; B)
;
;
;
;
; C)
;
;
;
;
; D)
;
;
;
;
; E)
;
;
;
;
; Unlock Deck
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36
Evaluate the function at
A)
B)
C)
D)
E)
at A)
B)
C)
D)
E)
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37
Sketch the level curves for the function below for the given c-values
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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38
Use the function to find
A)
B)
C)
D)
E)
to find A)
B)
C)
D)
E)
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39
A manufacturer estimates the Cobb-Douglas production function to be given by
Estimate the production levels when and
A) 135,540 units
B) 122,560 units
C) 131,601 units
D) 145,330 units
E) 112,745 units
Estimate the production levels when
and A) 135,540 units
B) 122,560 units
C) 131,601 units
D) 145,330 units
E) 112,745 units
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40
Describe the level curves of the function.Sketch the level curves for the given c-values. ,
A)
B)
C)
D)
E)
, A)
B)
C)
D)
E)
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41
The value of an investment of $18,000 after t years in an account for which the interest rate r is compounded continuously is given by the function dollars.Write the partial derivative
A)
B)
C)
D)
E)
of an investment of $18,000 after t years in an account for which the interest rate r is compounded continuously is given by the function dollars.Write the partial derivative A)
B)
C)
D)
E)
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42
Find the first partial derivatives with respect to x,y,and z.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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43
Evaluate and for the function at the point
A) and
B) and
C) and
D) and
E) and
and for the function at the point A)
and B)
and C)
and D)
and E)
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44
Find the four second partial derivatives.Observe that the second mixed partials are equal.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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45
Test for relative extrema and saddle points.
A) saddle point at
B) saddle point at
C) saddle point at
D) relative minimum at
E) relative minimum at
A) saddle point at
B) saddle point at
C) saddle point at
D) relative minimum at
E) relative minimum at
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46
Examine the function given below for relative extrema and saddle points.
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
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47
Examine the function given below for relative extrema and saddle points.
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a relative minimum at
B) The function has a relative maximum at
C) The function has a saddle point at
D) The function has a saddle point at
E) The function has a relative maximum at
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48
Examine the function for relative extrema.
A) saddle point at
B) relative minimum at
C) saddle point at
D) relative maximum at
E) saddle point at
for relative extrema. A) saddle point at
B) relative minimum at
C) saddle point at
D) relative maximum at
E) saddle point at
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49
Examine the function given below for relative extrema and saddle points.
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
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50
Examine the function given below for relative extrema and saddle points.
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
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51
Examine the function for relative extrema.
A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
for relative extrema. A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
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52
Evaluate and for the function at the point Round your answer to two decimal places.
A) and
B) and
C) and
D) and
E) and
and for the function at the point Round your answer to two decimal places. A)
and B)
and C)
and D)
and E)
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53
For ,find all values of x and y such that and simultaneously.
A)
B)
C) (0,0)
D) Both A and C
E) Both A and B
,find all values of x and y such that and simultaneously. A)
B)
C) (0,0)
D) Both A and C
E) Both A and B
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54
The utility function is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is Determine the marginal utility of product x.
A)
B)
C)
D)
E)
is a measure of utility (or satisfaction)derived by a person from the consumption of two products x and y.Suppose the utility function is Determine the marginal utility of product x. A)
B)
C)
D)
E)
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55
Examine the function given below for relative extrema and saddle points.
A) The function has a relative maximum at
B) The function has a relative minimum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
A) The function has a relative maximum at
B) The function has a relative minimum at
C) The function has a saddle point at
D) The function has a relative maximum at
E) The function has a relative minimum at
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56
Find the slopes of the surface in the x- and y- directions at the point
A)
Slope in x-direction: 1
Slope in y-direction: 5
B)
Slope in x-direction: -2
Slope in y-direction: 6
C)
Slope in x-direction: 5
Slope in y-direction: 1
D)
Slope in x-direction: 4
Slope in y-direction: 4
E)
Slope in x-direction: 6
Slope in y-direction: -2
in the x- and y- directions at the point A)
Slope in x-direction: 1
Slope in y-direction: 5
B)
Slope in x-direction: -2
Slope in y-direction: 6
C)
Slope in x-direction: 5
Slope in y-direction: 1
D)
Slope in x-direction: 4
Slope in y-direction: 4
E)
Slope in x-direction: 6
Slope in y-direction: -2
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57
Examine the function for relative extrema and saddle points.
A) saddle point: ; relative minimum:
B) relative minimum: ; relative maximum:
C) saddle points: ,
D) saddle point: ; relative minimum:
E) relative minimum: ; relative maximum:
for relative extrema and saddle points. A) saddle point:
; relative minimum: B) relative minimum:
; relative maximum: C) saddle points:
, D) saddle point:
; relative minimum: E) relative minimum:
; relative maximum: Unlock Deck
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58
Examine the function given below for relative extrema and saddle points.
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
A) The function has a saddle point at
B) The function has a relative maximum at
C) The function has a relative minimum at
D) The function has a saddle point at
E) The function has a relative maximum at
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59
Determine whether there is a relative maximum,a relative minimum,a saddle point,or insufficient information to determine the nature of the function at the critical point
A) insufficient information to determine the nature of the function at
B) relative maximum at
C) saddle point at
D) relative minimum at
at the critical point A) insufficient information to determine the nature of the function at
B) relative maximum at
C) saddle point at
D) relative minimum at
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60
A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model.The cost function for producing x freestanding and y fireplace-insert stoves is Find the marginal costs ( and )when and Round your answers to two decimal places.
A)
B)
C)
D)
E)
Find the marginal costs ( and )when and Round your answers to two decimal places. A)
B)
C)
D)
E)
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61
Find three positive numbers x,y,and z whose sum is 27 and product is a maximum.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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62
Use Lagrange multipliers to maximize the function subject to the following constraint: Assume that x,y,and z are positive.
A) 110
B)
C)
D) 10
E) no absolute maximum
subject to the following constraint: Assume that x,y,and z are positive.
A) 110
B)
C)
D) 10
E) no absolute maximum
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63
A rectangular box is resting on the xy-plane with one vertex at the origin.The opposite vertex lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize subject to the constraint ).
A) 15 units × 12 units × 5 units
B) 11 units × 9 units × 5 units
C) 10 units × 9 units × 6 units
D) 15 units × 10 units × 6 units
E) 12 units × 11 units × 7 units
Find the dimensions that maximize the volume. (Hint: Maximize subject to the constraint ).
A) 15 units × 12 units × 5 units
B) 11 units × 9 units × 5 units
C) 10 units × 9 units × 6 units
D) 15 units × 10 units × 6 units
E) 12 units × 11 units × 7 units
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64
Find the critical points of the function ,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point.
A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point. A) relative minimum at
B) relative maximum at
C) relative minimum at
D) relative maximum at
E) no relative extrema
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65
Use Lagrange multipliers to find the given extremum.In each case,assume that x and y are positive.
Maximize Constraint
A)
B)
C)
D)
E)
Maximize
Constraint A)
B)
C)
D)
E)
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66
A company manufactures two types of sneakers: running shoes and basketball shoes.The total revenue from x1 units of running shoes and x2 units of basketball shoes is: ,
Where x1 and x2 are in thousands of units.Find x1 and x2 so as to maximize the revenue.
A)
B)
C)
D)
E)
,
Where x1 and x2 are in thousands of units.Find x1 and x2 so as to maximize the revenue.
A)
B)
C)
D)
E)
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67
Find the critical points of the function ,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point.
A) relative minima at , , , where a, b, c, d, m, and n are arbitrary real numbers
B) relative maxima at , , , where a, b, c, d, m, and n are arbitrary real numbers
C) relative minimum at
D) relative maximum at
E) no relative extrema
,and,from the form of the function,determine whether a relative maximum or a relative minimum occurs at each point. A) relative minima at
, , , where a, b, c, d, m, and n are arbitrary real numbersB) relative maxima at
, , , where a, b, c, d, m, and n are arbitrary real numbersC) relative minimum at
D) relative maximum at
E) no relative extrema
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68
An agronomist used four test plots to determine the relationship between the wheat yield y (in bushels per acre)and the amount of fertilizer x (in pounds per acre).The results are shown in the table.
(I)Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data.
(II)Estimate the yield for a fertilizer application of 160 pounds per acre.
A) (I) ; (II) 39.10
B) (I) ; (II) 45.90
C) (I) ; (II) 44.18
D) (I) ; (II) 42.32
E) (I) ; (II)54.94
(I)Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data.
(II)Estimate the yield for a fertilizer application of 160 pounds per acre.
A) (I)
; (II) 39.10B) (I)
; (II) 45.90C) (I)
; (II) 44.18D) (I)
; (II) 42.32E) (I)
; (II)54.94 Unlock Deck
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69
Find the least squares regression line for the given points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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70
Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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71
A manufacturer has an order for 1,100 units of fine paper that can be produced at two locations.Let x1 and x2 be the numbers of units produced at the two plants.Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by
A) units and units
B) units and units
C) units and units
D) units and units
E) units and units
A)
units and unitsB)
units and unitsC)
units and unitsD)
units and unitsE)
units and units Unlock Deck
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72
Use Lagrange multipliers to find the given extremum.In each case,assume that x,y,and z are positive.
Maximize Constraints
A)
B)
C)
D)
E)
Maximize
Constraints A)
B)
C)
D)
E)
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73
The sum of the length (denote by z)and the girth (perimeter of a cross section)of packages carried by a delivery service cannot exceed 36 inches.Find the dimensions of the rectangular package of largest volume that may be sent.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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74
Find the sum of the squared errors for the linear model and the quadratic model using the given points.
A)
B)
C)
D)
E)
and the quadratic model using the given points. A)
B)
C)
D)
E)
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75
Use Lagrange multipliers to find the given extremum.Assume that x and y are positive.
Maximize Constraint
A)
B)
C)
D)
E)
Maximize
Constraint A)
B)
C)
D)
E)
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76
Use Lagrange multipliers to minimize the function subject to the following constraint.
Assume that x,y,and z are positive.
A) 4
B) 12
C) 8
D) 24
E) 36
subject to the following constraint. Assume that x,y,and z are positive.
A) 4
B) 12
C) 8
D) 24
E) 36
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77
A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals.Fencing for the perimeter costs 10 per foot.To separate the corrals,a fence that costs 8 per foot will divide the region.The total area of the two corrals is to be 7,000 square feet.Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing.
A) dimensions: 50 feet by 70 feet
B) dimensions: 140 feet by 50 feet
C) dimensions: 25 feet by 70 feet
D) dimensions: 35 feet by 50 feet
E) dimensions: 25 feet by 35 feet
A) dimensions: 50 feet by 70 feet
B) dimensions: 140 feet by 50 feet
C) dimensions: 25 feet by 70 feet
D) dimensions: 35 feet by 50 feet
E) dimensions: 25 feet by 35 feet
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78
A microbiologist must prepare a culture medium in which to grow a certain type of bacteria.The percent of salt contained in this medium is given by
Where x,y,and z are the nutrient solutions to be mixed in the medium.For the bacteria to grow,the medium must be 13% salt.Nutrient solutions x,y,and z cost $1,$2,and $3 per liter,respectively.How much of each nutrient solution should be used to minimize the cost of the culture medium?
A)
B)
C)
D)
E)
Where x,y,and z are the nutrient solutions to be mixed in the medium.For the bacteria to grow,the medium must be 13% salt.Nutrient solutions x,y,and z cost $1,$2,and $3 per liter,respectively.How much of each nutrient solution should be used to minimize the cost of the culture medium?
A)
B)
C)
D)
E)
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79
Use Lagrange multipliers to find the minimum distance from the circle to the point Round your answer to the nearest tenth.
A) 13.2
B) 160.0
C) 173.7
D) 10.8
E) 116.8
to the point Round your answer to the nearest tenth. A) 13.2
B) 160.0
C) 173.7
D) 10.8
E) 116.8
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80
Find three positive numbers x,y,and z whose sum is 18 and the sum of the squares is a minimum.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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