Deck 6: Applications of Integration

Question 1

Let be the probability density function of a random variable X. Find the median of the probability density function .

A) e.3
B) f.6
C) g.2
D)1
H)0

Answer 1

C

Question 2

Find c so that the following can serve as the probability density function of a random variable X: a.8
E)1
B)4
F)16
C) g. d. h.2

Answer 2

A

Question 3

Assume the daily consumption of electric power (in millions of kilowatt-hours) of a certain city
has the probability density If the city's power plant has a daily capacity of 12 million kilowatt-hours, what is the probability that the available power supply will be inadequate on any given day?

Answer 3

Question 4

Let X be a continuous random variable with density function If the median of this distribution is , then c is:

A) e.3
B) f. c.2 g. d. h.3

Question 5

IQ scores are assumed to be normally distributed with a mean and standard deviation . Use either Simpson's Rule or the Midpoint Rule to approximate the probability that a person selected at random from the general population will have an IQ score
(a) between 70 and 130.(b) over 130.

Question 6

(a) Explain why the function defined by the graph below is a probability density function. (b) Use the graph to find the following probabilities:
(i) (ii) (c) Calculate the median for this distribution.

Question 7

Let be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is mean of the probability density function?

Question 8

(a) Show that where k is fixed and is a probability density function.(b) What is the mean for this distribution?
(c) Calculate the median of

Question 9

Find k so that the function can serve as the probability density function of a random variable X.

A) e.3
B) f.6
C) g.2
D)1
H)0

Question 10

Let X be a continuous random variable with density function What is the mean of X?:

A) e. b.1
F) c. g. d. h.

Question 11

Find k so that the function can serve as the probability density function of a random variable X.

A) e.3
B) f.6
C) g.2
D)1
H)0

Question 12

Let be the probability density function of a random variable X. Find the median of the probability density function .

A) e. b. f. c. g. d. h.

Question 13

Assume the weights of adult males are normally distributed with a mean weight of 150 pounds and a standard deviation of 20 pounds. Use Simpson's Rule or the Midpoint Rule to estimate the following:
(a) What is the probability that an adult male chosen at random will weigh between 120 pounds and 180 pounds?
(b) What percentage of the adult male population weighs more than 200 pounds?

Question 14

Let be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is median of the probability density function?

Question 15

Suppose that the mileage (in thousands of miles) which car owners can obtain from a certain kind of tire has the probability density Find the probability that a tire chosen at random will last
(a) at most 10,000 miles.(b) between 15,000 and 25,000 miles.(c) at least 30,000 miles.

Question 16

The density function for the life of a certain type of battery is modeled by and is measured in months.(a) What is the probability that a battery will wear out during the first month of use?
(b) What is the probability that a battery is functioning for more than 12 months?

Question 17

Let be the probability density function of a random variable X. Find the mean of the probability density function .

A) e.3
B) f.6
C) g.2
D)1
H)0

Question 18

(a) Show that is the probability density function of a random variable.(b) What is the mean for this distribution?
(c) Calculate the median of

Question 19

Let be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is the probability that a customer will wait more than 10 minutes?

Question 20

Let (a) Find k so that f can serve as the probability density function of a random variable X.(b) Find .(c) Find the mean.

Question 21

The marginal revenue for a company when sales are q units is given by . Find the increase in revenue when the sales level increases from 100 to 200 units.

Question 22

A manufacture has been selling 1000 ceiling fans at $60 each. A market survey indicates that for every $10 that price is reduced, the number of sets sold will increase by 100. Find the demand function and calculate the consumer surplus when the selling price is set at $50.

Question 23

The demand function for a certain commodity is . Find the consumer surplus when the selling price is $18.

Question 24

The density function for the waiting time at a bank is modeled by and is measured in minutes.(a) Find the median waiting time.(b) Find the mean waiting time.(c) Sketch the graph of the density function showing the median and mean.

Question 25

A supply function is given by , where x is the number of units produced. Find the producer surplus when the selling price is $15.

A)100
E)800
B)500
F)400
C)250
G)200
D)1000
H)2000

Question 26

The marginal cost for a company is given by where q is the number of units produced. What is the total cost to raise production from 100 to 200 units?

Question 27

A rental estate management company manages an apartment complex with 20 units. The manager estimates that all 20 units can be rented if the rent is $150 per unit per month and that for each increase in rent of $10, one apartment will be vacated. Find the consumer surplus when 5 apartments are vacated.

A)3000
E)1125
B)2250
F)6000
C)1250
G)2500
D)1000
H)3500

Question 28

The demand function for a certain commodity is Find the consumer's surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer's surplus as an area.

Question 29

The density function for the life of a certain type of battery is modeled by and is measured in months.(a) Find the median life of the batteries.(b) Find the mean life of the batteries.(c) Sketch the graph of the density function showing the median and mean.

Question 30

The demand function for a certain commodity is . Find the consumer surplus when the selling price is $18?

A)135
E)270
B)30
F)40
C)45
G)90
D)360
H)180

Question 31

The demand function for a certain commodity is Find the consumer surplus when the sales level is 30.

A)90
E)30
B)45
F)60
C)15
G)20
D)70
H)80

Question 32

The demand function for producing a certain commodity is given by . Find the consumer surplus when the sale level is 500.

Question 33

The marginal cost for the production of the new Super Widget at Widget International is give by , whereas the marginal revenue is , where in both cases q represents the number of units produced.(a) Determine the change in profits when sales are increased from 700 to 1700 units.(b) What is the change in profit when sales increase from 1700 to 2700 units? Discuss your answer.

Question 34

The density function for the waiting time at a bank is modeled by and is measured in minutes.(a) What is the probability that a customer will be served within the first 5 minutes?
(b) What is the probability that a customer has to wait for more than 15 minutes?

Question 35

If the demand function for a certain commodity is and the consumer surplus is 15, what should be the sale price?

A)1
E)3.3
B)4
F)6
C)1.5
G)2
D)2.5
H)3

Question 36

The demand function for a certain commodity is , and consumer surplus is 90. What should be the selling price?

Question 37

If the demand function for a certain commodity is and the consumer surplus is 15, what should be the production level?

A)90
E)30
B)45
F)60
C)15
G)20
D)70
H)80

Question 38

Suppose a company has estimated that the marginal cost of manufacturing x pairs of a new line of jeans is (measured in dollars per pair) with a fixed start-up cost of c(0) = 2000. Find the cost of producing the first 1000 pair of jeans.

A)$2000
E)$3000
B)$4000
F)$5000
C)$6000
G)$8000
D)$10,000
H)$15,000

Question 39

Suppose a company has estimated that the marginal cost of manufacturing x items is (measured in dollars per unit) with a fixed start-up cost of c(0) = 10,000. Find the cost of producing the first 500 items.

A)$15,000
E)$25,000
B)$5,000
F)$6000
C)$10,000
G)$20,000
D)$17,500
H)$60,000

Question 40

The marginal revenue from selling x items is . The revenue from the sale of the first 100 items is $8800. What is the revenue from the sale of the first 200 items?

Question 41

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft³ . Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

A)336
E)60
B)30
F)120
C)168
G)28
D)240
H)12

Question 42

Find the y-coordinate of the centroid of the region bounded by the curves a.0.85
E)0.75
B)0.70
F)0.55
C)0.60
G)0.80
D)0.50
H)0.65

Question 43

Find the center of mass of the linear system a.9
E) b. f. c. g. d.0
H)18

Question 44

Find the center of mass of the system a. e. b. f. c. g. d. h.

Question 45

A force of 8 dynes is required to stretch a spring from its natural length of 10 cm to a length of 15 cm. How much work is done
(a) in stretching the spring to a length of 25 cm?
(b) in stretching the spring from a length of 20 cm to a length of 25 cm?

Question 46

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?

A)8 e.5 b.24 f.4 c.16 g.30 d.6 h.18

Question 47

A swimming pool 24 feet long and 15 feet wide has a bottom that is an inclined plane, the shallow end having a depth of 3 feet, and the deep end 10 feet. The pool is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pounds on the bottom of the pool.

A)145,750
E)146,500
B)146,250
F)147,500
C)147,000
G)146,750
D)147,250
H)146,000

Question 48

Find the center of mass of the linear system .

A) e. b. f. c.6
G) d. h.

Question 49

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, find the hydrostatic force in pounds on the side of the tank.

A)30 e.30
B)60
F)120 c.240 g.60 d.240
H)120

Question 50

A rope 100 feet long weighing 2 pounds per foot hangs over the edge of a building 100 feet tall. How much work in foot-pounds is done in pulling the rope to the top of the building?

A)7500
E)10,000
B)5000
F)750
C)1000
G)1250
D)500
H)12,500

Question 51

The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by , where t is measured in seconds. Find the cardiac output.

Question 52

The following table shows the relationship between price and demand for milk produced in a large dairy.q (billions of pounds of milk per year)
45
50
55
60
65
70
75
p (price in dollars per pound)
1.00
0.90
0.80
0.70
0.60
0.50
0.40
Determine the consumer's surplus when the sales quality is 65 billion pounds of milk in a year. Illustrate your answer by drawing the corresponding demand curve and the identifying the consumer's surplus as a region.

Question 53

Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft.

A) e.10⁶
B) f. c. g. d. h.

Question 54

A force of 20 pounds stretches a spring foot beyond its natural length. Find the work done in stretching the spring 1 foot.

A)20 ft-lb
E) ft-lb
B) ft-lb
F) ft-lb
C) ft-lb
G) ft-lb
D) ft-lb
H) ft-lb

Question 55

Find the x-coordinate of the centroid of the region bounded by the graphs and the x-axis.

A)8
E)4
B) f. c. g. d. h.

Question 56

If ft-lb of work is needed to stretch a spring a length ft beyond its natural length, find the spring constant.

Question 57

A spring stretches 1 foot beyond its natural position under a force of 100 points. How much work in foot-pounds is done in stretching it 3 feet beyond its natural position?

A)600
E)100
B)30
F)900
C)1500
G)150
D)450
H)300

Question 58

A gate in an irrigation canal is in the form of a trapezoid 3 feet wide at the bottom, 5 feet wide at the top, with height equal to 2 feet. It is placed vertically in the canal with the water extending to its top. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pounds on the gate.

A)360
E)400
B)380
F)460
C)440
G)500
D)420
H)480

Question 59

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines a. e. b. f. c. g. d. h.

Question 60

A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?

Question 61

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Question 62

A 100-foot length of steel chain weighing 5 lb/ft is dangling from the drum of a winch.(a) How much work is required to wind the chain onto the drum?
(b) How much work is required to wind the first 30 feet of chain onto the drum?

Question 63

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Question 64

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Question 65

The tank in the figure below is full of water with a density 62.5 lb/ft³. How much work is required to empty the tank by pumping water to a point 4 feet above the top of the tank?

Question 66

A tank 5 feet long has cross-sections n the shape of a parabola (where x and y are in feet). Suppose that the tank is filled to a depth of 3 feet with liquid weighing 15 lb/ft³. How much work is required to empty the tank by pumping the liquid over the edge of the tank?

Question 67

How much work is done to bring a load of ore to the surface if a miner uses a cable weighing 2 lb/ft to haul a 100 lb bucket of ore up a mine shaft 800 feet deep?

Question 68

Determine the centroid of the region bounded by the equation in the first quadrant between x = 1 and x = 4.

Question 69

Find the total hydraulic force on a dam in the shape of an equilateral triangle with one vertex pointing down, if the side of the triangle is 100 feet and the water is even with the top.

Question 70

Find the centroid of the region bounded by .

Question 71

City B (population 4,000) is 10 miles north of City A (population 6,000), and City C (population 5,000) is 30 miles east of City B. Where is the best place to locate a super market serving the people in these cities?

Question 72

A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m³ to a depth of 2.5 m. Find
(a) the hydrostatic pressure at the bottom of the pool.(b) the hydrostatic force on the bottom.(c) the hydrostatic force on one end of the pool.

Question 73

A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.

Question 74

A flat plate of negligible thickness is in the shape of a right triangle with base 5 ft and height 10 ft. The plate is submerged in a tank of water. Find the force on the face of the plate under the following conditions: (Use 62.4 lb/ft³ as the density of water.)
(a) The plate is submerged horizontally so that it rests flat on the bottom of the tank at a depth of 14 ft.(b) The plate is submerged vertically, base edge up and base at a depth of 3 ft.

Question 75

Calculate the center of mass of a lamina with the given density and shape:

Question 76

Find the center of mass of the lamina of uniform density bounded by and x-axis.

Question 77

Find the centroid of the region bounded by

Question 78

Find the centroid of the region bounded by .

Question 79

Suppose a hemispherical tank with radius 10 feet is filled with liquid whose density is 62 pounds per cubic foot. Find the work required to pump all the liquid out through the top of the tank.

Question 80

Two people weighing 100 lb and 180 lb are at opposite ends of a seesaw 14 ft long. Where should the point of support be placed so that the seesaw will balance?