Question 1

When pumping water out of a full hemispherical basin with radius r feet, how far down will the water level be when 1/4 of the work has been done? [The density of water is 62.4 lbs/ft3.]

Accepted Answer

A)   
B)   
C)   
D)

Question 2

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral  .

Accepted Answer

A)   
B)   
C)   
D)

Question 3

What is the maximum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 50 feet? [The density of water is 62.4 lbs/ft3.]

Accepted Answer

A)  5,200,000 lbs 
B)  7,800,000 lbs 
C)  10,400,000 lbs 
D)  11,031 lbs

Question 4

An object is dropped from a height of 140 feet. Another object directly below the first is launched vertically from the ground with an initial velocity of 70 ft/s. Determine when and how high up the objects collide.

Accepted Answer

The answer of An object is dropped from a height...

Question 5

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

Accepted Answer

A)  y(0) = 30, y'(0) = -6 
B)  y(0) = 0, y'(0) = -6 
C)  y(0) = 30, y'(0) = 6 
D)  y(0) = 0, y'(0) = 6

Question 6

An underwater viewing window is rectangular with width 40 feet. The window extends from the surface of the water to a depth of 20 feet. Find the hydrostatic force on the window.

Accepted Answer

A)  24,960 lb 
B)  998,400 lb 
C)  49,920 lb 
D)  499,200 lb

Question 7

Let R be the region bounded by  , the x-axis, and the y-axis. Compute the volume of the solid formed by revolving R about the y-axis.

Accepted Answer

A)   
B)   
C)   
D)

Question 8

A force of 7 pounds stretches a spring 7 inches. Find the work done in stretching this spring 10 inches beyond its natural length.

Accepted Answer

A)   foot-pounds 
B)   foot-pounds 
C)   foot-pounds 
D)   foot-pounds

Question 9

A cylindrical water tank has a radius of 4 feet and a height of 3.0 feet. Compute the work done to pump the water out of a filled tank through the top. [The density of water is 62.4 lbs/ft3.]

Accepted Answer

A)  14,115 ft-lbs 
B)  4705 ft-lbs 
C)  281 ft-lbs 
D)  7390 ft-lbs

Question 10

A projectile traveling with momentum 4 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s2 and it is exerted over a time range  . What is the object's momentum (in the direction of the impulse) after the impulse?

Accepted Answer

A)  8 kg m/s 
B)  -20 kg m/s 
C)  -19 kg m/s 
D)  16 kg m/s

Question 11

Compute the volume of the solid formed by revolving the region bounded by  and  about the x-axis.

Accepted Answer

A)   
B)   
C)   
D)

Question 12

A 2-person tent has a rectangular base  and rises to a single point 5' up. Compute the volume of the tent.  7 ft 8 ft

Accepted Answer

A)  75 ft3 
B)  70 ft3 
C)  140 ft3 
D)  280 ft3

Question 13

Identify the graph and area of the region bounded by the curves  . The figures below are not necessarily to scale.

Accepted Answer

A)  area =   
B)  area =   
C)  area =   
D)  area =

Question 14

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Accepted Answer

A)  y(0) = 0, y'(0) = 0 
B)  y(0) = 45, y'(0) = 45t 
C)  y(0) = 0, y'(0) = -45 
D)  y(0) = 45, y'(0) = 0

Question 15

Where is the center of mass of a region of uniform density bounded by  ?

Accepted Answer

A)   
B)   
C)   
D)

Question 16

Compute the arc length exactly.  , -2 $\leq$ x $\leq$ -1

Accepted Answer

A)   
B)   
C)   
D)

Question 17

An object propelled from the ground with an initial velocity of 95 ft/s will reach a maximum height of 141.0 ft. If the initial velocity is increased 13%, by what percentage will the maximum height increase? Round percentages to the nearest integer.

Accepted Answer

A)  13% 
B)  28% 
C)  2% 
D)  22%

Question 18

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral  .

Accepted Answer

A)   
B)   
C)   
D)

Question 19

Given a certain amount of land potentially devoted to farming, one will first use the more fertile regions where production costs are lowest before using the less fertile regions. The consequence of this logical strategy is that the production costs increase as production increases. A certain landowner has enough land to produce 106 bushels of a particular crop, and the production costs, c, of the nth bushel is described with the equation  . If the market price is p for this particular crop, the profit for producing the nth bushel is  . If p = $5.25, how much total profit could the farmer expect to obtain by producing  bushels.

Accepted Answer

A)  $377,600 
B)  $289,200 
C)  $350,000 
D)  $322,400

Question 20

The base of a solid V is the region bounded by  and  Find the volume if V has semicircular cross sections perpendicular to the x-axis.

Accepted Answer

A)   
B)   
C)   
D)

Exam 6: Applications of the Definite Integral

When pumping water out of a full hemispherical basin with radius r feet, how far down will the water level be when 1/4 of the work has been done? [The density of water is 62.4 lbs/ft³.]

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

What is the maximum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 50 feet? [The density of water is 62.4 lbs/ft³.]

An object is dropped from a height of 140 feet. Another object directly below the first is launched vertically from the ground with an initial velocity of 70 ft/s. Determine when and how high up the objects collide.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

An underwater viewing window is rectangular with width 40 feet. The window extends from the surface of the water to a depth of 20 feet. Find the hydrostatic force on the window.

Let R be the region bounded by , the x-axis, and the y-axis. Compute the volume of the solid formed by revolving R about the y-axis.

A force of 7 pounds stretches a spring 7 inches. Find the work done in stretching this spring 10 inches beyond its natural length.

A cylindrical water tank has a radius of 4 feet and a height of 3.0 feet. Compute the work done to pump the water out of a filled tank through the top. [The density of water is 62.4 lbs/ft³.]

A projectile traveling with momentum 4 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s² and it is exerted over a time range . What is the object's momentum (in the direction of the impulse) after the impulse?

Compute the volume of the solid formed by revolving the region bounded by and about the x-axis.

A 2-person tent has a rectangular base and rises to a single point 5' up. Compute the volume of the tent. 7 ft 8 ft

Identify the graph and area of the region bounded by the curves . The figures below are not necessarily to scale.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Where is the center of mass of a region of uniform density bounded by ?

Compute the arc length exactly. $Compute the arc length exactly. , -2 \leq x \leq -1 A) B) C) D)$ , -2 $\leq$ x $\leq$ -1

An object propelled from the ground with an initial velocity of 95 ft/s will reach a maximum height of 141.0 ft. If the initial velocity is increased 13%, by what percentage will the maximum height increase? Round percentages to the nearest integer.

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

The base of a solid V is the region bounded by and Find the volume if V has semicircular cross sections perpendicular to the x-axis.

Exam 6: Applications of the Definite Integral

When pumping water out of a full hemispherical basin with radius r feet, how far down will the water level be when 1/4 of the work has been done? [The density of water is 62.4 lbs/ft3.]

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

What is the maximum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 50 feet? [The density of water is 62.4 lbs/ft3.]

An object is dropped from a height of 140 feet. Another object directly below the first is launched vertically from the ground with an initial velocity of 70 ft/s. Determine when and how high up the objects collide.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

An underwater viewing window is rectangular with width 40 feet. The window extends from the surface of the water to a depth of 20 feet. Find the hydrostatic force on the window.

Let R be the region bounded by , the x-axis, and the y-axis. Compute the volume of the solid formed by revolving R about the y-axis.

A force of 7 pounds stretches a spring 7 inches. Find the work done in stretching this spring 10 inches beyond its natural length.

A cylindrical water tank has a radius of 4 feet and a height of 3.0 feet. Compute the work done to pump the water out of a filled tank through the top. [The density of water is 62.4 lbs/ft3.]

A projectile traveling with momentum 4 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s2 and it is exerted over a time range . What is the object's momentum (in the direction of the impulse) after the impulse?

Compute the volume of the solid formed by revolving the region bounded by and about the x-axis.

A 2-person tent has a rectangular base and rises to a single point 5' up. Compute the volume of the tent. 7 ft 8 ft

Identify the graph and area of the region bounded by the curves . The figures below are not necessarily to scale.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Where is the center of mass of a region of uniform density bounded by ?

Compute the arc length exactly. , -2 ≤\leq≤ x ≤\leq≤ -1

An object propelled from the ground with an initial velocity of 95 ft/s will reach a maximum height of 141.0 ft. If the initial velocity is increased 13%, by what percentage will the maximum height increase? Round percentages to the nearest integer.

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

The base of a solid V is the region bounded by and Find the volume if V has semicircular cross sections perpendicular to the x-axis.

When pumping water out of a full hemispherical basin with radius r feet, how far down will the water level be when 1/4 of the work has been done? [The density of water is 62.4 lbs/ft³.]

What is the maximum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 50 feet? [The density of water is 62.4 lbs/ft³.]

A cylindrical water tank has a radius of 4 feet and a height of 3.0 feet. Compute the work done to pump the water out of a filled tank through the top. [The density of water is 62.4 lbs/ft³.]

A projectile traveling with momentum 4 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s² and it is exerted over a time range . What is the object's momentum (in the direction of the impulse) after the impulse?

Compute the arc length exactly. $Compute the arc length exactly. , -2 \leq x \leq -1 A) B) C) D)$ , -2 $\leq$ x $\leq$ -1