Question 1

A study of the costs to produce airplanes in World War II led to the theory of "learning curves," the idea of which is that the marginal cost per plane decreases over the duration of a production run.In other words, with experience, staff on an assembly line can produce planes with greater efficiency.The 90% learning curve describes a typical situation where the marginal cost, MC, to produce the x^th plane is given by $M C(x)=M M_{0} x^{\log _{2} 0.9}$ , where $M_{0}$ = marginal cost to produce the first plane.[Note: You may use the fact that $\log _{2} x=\frac{\ln x}{\ln 2}$ .] If a plant produces planes with a 90% learning curve on production costs, and the marginal cost for the first plane is $700,000, then what is the marginal cost to produce the fourth plane? Round to the nearest thousand dollars.

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Question 2

An object is in the shape drawn below; its boundary is obtained by rotating the parabola $y=2 x^{2}$ (for 0 $\le$x$\le$ 1)around the y axis.(Units are in centimeters.)Suppose that the density of this object varies with height according to the rule $\rho$(y)= 12 (2 - y)grams/cm³.Which of the following Riemann sums computes (approximately)the mass in grams of this object?

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Question 3

The work done lifting a 5-lb bag of sugar 4 feet off the floor is _____ ft-lbs.

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Question 4

If it is known to take 147 joules of work to lift a box 1.5 meters off the floor, what is the mass of the box in kg?

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Question 5

A swimming pool has shape as shown in the following figure. If the pool is 9 ft deep, how many foot-pounds does it take to pump all the water out? (Note: water weighs 62.4 pounds/ft³.)Round to the nearest whole number.

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Question 6

The density of cars (in cars per mile)down a 24-mile stretch of the Massachusetts Turnpike starting at a toll plaza is given by $\rho(x)=500+100 \sin (\pi x)$ , where x is the distance in miles from the toll plaza and 0 $\le$ x $\le$ 24.Estimate the total number of cars down the 24-mile stretch by creating a Reimann sum, converting it to an integral, and evaluating it.

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Question 7

A bridge worker needs to pull a 14-meter uniform cable with mass 3 kg/meter up to the work platform.How many joules are needed, assuming that the cable is hanging straight down from the platform? Round to the nearest whole number.

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Question 8

Circle City is circular with a radius of 3 miles.Right in the center is a circular park with radius one mile.No one lives in the park.Elsewhere the population density is 3000(5 - r)people per square mile, where r is the distance from the center in miles.What is the total population of Circle City, to the nearest thousand?

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Question 9

A flat metal plate is in the shape determined by the area under the graph of $f(x)=\frac{1}{1+x}$ between x = 0 and x = 4.The density of the plate x units from the y-axis is given by x² grams/cm².Write down a Riemann sum which approximates the total mass.Convert it to an integral, and evaluate the integral to 3 decimal places.

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Question 10

A study of the costs to produce airplanes in World War II led to the theory of "learning curves," the idea of which is that the marginal cost per plane decreases over the duration of a production run.In other words, with experience, staff on an assembly line can produce planes with greater efficiency.The 90% learning curve describes a typical situation where the marginal cost, MC, to produce the x^th plane is given by $M C(x)=M M_{0} x^{\log _{2} 0.9}$ , where $M_{0}$ = marginal cost to produce the first plane.[Note: You may use the fact that $\log _{2} x=\frac{\ln x}{\ln 2}$ .] Recall that marginal cost is related to total cost as follows: $M C(x)=C^{\prime}(x)$ , where C(x)= total cost to produce x units. Given this, and the formula for MC(x)with $M_{0}$ = $500,000, find a formula for C(x).What, physically, is the meaning of the constant of integration in your formula?

Accepted Answer

The constant of integration in the context of integrating a marginal cost function to find the total cost function represents the initial fixed costs or setup costs before any units are produced. This is not the cost of producing the first unit, but rather the costs incurred to prepare for production, such as setting up the plant or assembly line.

Question 11

The force of gravitational attraction between a thin rod of mass M and length L and a particle of mass m lying on the same line as the rod at a distance of A from one of the ends is $\frac{G m M}{A(L+A)}$ .Use this result to select an integral for the total force due to gravity between two thin rods, both of mass M and length L, lying along the same line and separated by a distance A.[Hint: Divide one of the rods into small pieces, each of length dx and mass $\frac{M}{L} d x$ .Apply the formula above to each of the pieces, then form a Riemann sum.You know the rest…]

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Question 12

An object is in the shape drawn below; its boundary is obtained by rotating the parabola $y=2 x^{2}$ (for 0$\le$x $\le$ 1)around the y axis.(Units are in centimeters.)Suppose that the density of this object varies with height according to the rule $\rho$(y)= 8 (2 - y)grams/cm³.Compute the mass in grams of this object.Give an exact answer and then round it to 2 decimal places.

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Question 13

A 3-gram drop of thick red paint is added to a large can of white paint.A red disk forms and spreads outward, growing lighter at the edges.Since the amount of red paint stays constant through time, the density of the red paint in the disk must vary with time.Suppose that its density p in gm/cm² is of the form $p=k(t) f(r)$ for some functions k(t)of time and f(r)of the distance to the center of the disk.Let R(t)be the radius of the disk at time t.Which of the following equations expresses the fact that there are 3 grams of red paint in the disk?

Accepted Answer

The total amount of red paint is constant at 3 grams, which is represented by integrating the density function $p=k(t)f(r)$ over the area of the disk. The correct equation, $3=k(t)\int_{0}^{R(t)}f(r) 2 \pi r dr$, ensures that when you integrate the density function over the entire area of the disk, you get the total mass of the red paint, which is 3 grams.

Question 14

The Great Cone of Haverford College is a monument built by freshmen during a customs week long, long ago.It is 100 ft.high and its base has a diameter of 100 ft.It has been built from bricks (purportedly made of straw)which weigh 2 lbs/ft³.Use a definite integral to approximate the number of foot-pounds required to build the Cone.Round to the nearest million.

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Question 15

The density of cars (in cars per mile)down a 20-mile stretch of the Massachusetts Turnpike starting at a toll plaza is given by $\rho(x)=500+100 \sin (\pi x)$ , where x is the distance in miles from the toll plaza and 0 $\le$ x $\le$ 20.Which of the following Riemann sums estimates the total number of cars down the 20-mile stretch?

Accepted Answer

The correct Riemann sum formula to estimate the total number of cars involves multiplying the density function $\rho(x_i)$ by the width of each segment $\Delta x$, and summing over all segments. This is accurately represented in option D, where the 20-mile stretch is divided into $n$ segments, each of length $\frac{20}{n} = \Delta x$, and $x_i$ is a point in the $i^{th}$ segment.

Question 16

Circle City is circular with a radius of 3 miles.Right in the center is a circular park with radius one mile.No one lives in the park.Elsewhere the population density is 4500(5 - r)people per square mile, where r is the distance from the center in miles.What is the average population density of Circle City, to the nearest whole number?

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Question 17

A 5-gram drop of thick red paint is added to a large can of white paint.A red disk forms and spreads outward, growing lighter at the edges.Since the amount of red paint stays constant through time, the density of the red paint in the disk must vary with time.Suppose that its density p in gm/cm² is of the form $p=k(t) f(r)$ for some functions k(t)of time and f(r)of the distance to the center of the disk.For fixed r, which of the following integrals gives the average density of red paint at a distance r from the center of the disk from 0 to T seconds?

Accepted Answer

The average density of red paint at a distance r from the center over time is obtained by averaging the time-dependent part of the density function, $k(t)$, over the interval from 0 to T seconds. Since $p=k(t)f(r)$, and we are looking for the average density at a fixed r, we integrate $k(t)$ over time and divide by T, while $f(r)$ remains constant for the fixed distance r. This is correctly represented by $\frac{f(r)}{T} \int_{0}^{T} k(t) d t$.

Question 18

A chlorine solution is poured over the surface of a rectangular swimming pool that is 20 meters long, 10 meters wide and 2 meters deep everywhere.Before the circulating pumps in the pool are turned on, it is discovered that the density of the chlorine solution at a height h meters above the bottom of the pool is given by $\rho$(h)= 100h grams/m^3. In other words, the chlorine solution has distributed itself so that its density increases linearly from the bottom of the pool.Write a Riemann sum that approximates the total mass of chlorine solution in the pool, convert it into a definite integral, and evaluate it to find the total mass of the chlorine solution in the pool.Give your answer in kg.

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Question 19

A bridge worker needs to pull a 10-meter uniform cable with mass 3 kg/meter up to the work platform.A second worker needs to pull an identical cable (which is one-half as long)up one-half the original distance.Will the second worker do one-half the work of the first worker? Assume that each cable is hanging straight down from its platform.

Accepted Answer

The work done to lift a cable is directly proportional to both the mass of the cable and the height it is lifted. Since the second cable is half as long and only lifted half the distance, its mass is half that of the first cable, and it is lifted half as far. However, work is calculated as force times distance. The force needed is directly proportional to the mass, so halving the mass and the distance results in one-quarter the work done, not half.

Question 20

The density of a compressible liquid is 30(5 - h)kg/m³ at a height of h meters above the bottom.The liquid is put in the container as shown below (resting on the triangular side).The cross sections of the container are isosceles triangles, it has straight sides, and it looks like a triangular prism.How many kg will it hold? Round to the nearest whole number.

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Quiz 8: Using the Definite Integral

The work done lifting a 5-lb bag of sugar 4 feet off the floor is _____ ft-lbs.

If it is known to take 147 joules of work to lift a box 1.5 meters off the floor, what is the mass of the box in kg?

A swimming pool has shape as shown in the following figure. If the pool is 9 ft deep, how many foot-pounds does it take to pump all the water out? (Note: water weighs 62.4 pounds/ft3.)Round to the nearest whole number.

A bridge worker needs to pull a 14-meter uniform cable with mass 3 kg/meter up to the work platform.How many joules are needed, assuming that the cable is hanging straight down from the platform? Round to the nearest whole number.

A swimming pool has shape as shown in the following figure. If the pool is 9 ft deep, how many foot-pounds does it take to pump all the water out? (Note: water weighs 62.4 pounds/ft³.)Round to the nearest whole number.