Chapter 5

VOLUME OF SOLID OF REVOLUTION In Exercises 55 through 58 , find the volume of the solid of revolution formed by rotating the specified region \(R\) about the \(x\) axis. \(R\) is the region under the curve \(y=\frac{x+1}{\sqrt{x}}\) from \(x=1\) to \(x=4\).

In Exercises 59 through 62, solve the given initial value problem. \(\frac{d y}{d x}=2\), where \(y=4\) when \(x=-3\)

In Exercises 59 through 62, solve the given initial value problem. \(\frac{d y}{d x}=x(x-1)\), where \(y=1\) when \(x=1\)

In Exercises 59 through 62, solve the given initial value problem. \(\frac{d x}{d t}=e^{-2 t}\), where \(x=4\) when \(t=0\)

In Exercises 59 through 62, solve the given initial value problem. \(\frac{d y}{d t}=\frac{t+1}{t}\), where \(y=3\) when \(t=1\)

Find the function whose tangent line has slope \(x\left(x^{2}+1\right)^{-1}\) for each \(x\) and whose graph passes through the point \((1,5)\).

Find the function whose tangent line has slope \(x e^{-2 x^{2}}\) for each \(x\) and whose graph passes through the point \((0,-3)\).

NET ASSET VALUE It is estimated that \(t\) days from now a farmer's crop will be increasing at the rate of \(0.5 t^{2}+4(t+1)^{-1}\) bushels per day. By how much will the value of the crop increase during the next 6 days if the market price remains fixed at \(\$ 2\) per bushel?

DEPRECIATION The resale value of a certain industrial machine decreases at a rate that changes with time. When the machine is \(t\) years old, the rate at which its value is changing is \(200(t-6)\) dollars per year. If the machine was bought new for \(\$ 12,000\), how much will it be worth 10 years later?

TICKET SALES The promoters of a county fair estimate that \(t\) hours after the gates open at \(9: 00\) A.M. visitors will be entering the fair at the rate of \(-4(t+2)^{3}+54(t+2)^{2}\) people per hour. How many people will enter the fair between 10:00 A.M. and noon?

MARGINAL COST At a certain factory, the marginal cost is \(6(q-5)^{2}\) dollars per unit when the level of production is \(q\) units. By how much will the total manufacturing cost increase if the level of production is raised from 10 to 13 units?

PUBLIC TRANSPORTATION It is estimated that \(x\) weeks from now, the number of commuters using a new subway line will be increasing at the rate of \(18 x^{2}+500\) per week. Currently, 8,000 commuters use the subway. How many will be using it 5 weeks from now?

NET CHANGE IN BIOMASS A protein with mass \(m\) (grams) disintegrates into amino acids at a rate given by $$ \frac{d m}{d t}=\frac{-15 t}{t^{2}+5} $$ What is the net change in mass of the protein during the first 4 hours?

CONSUMPTION OF OIL It is estimated that \(t\) years from the beginning of the year 2012 , the demand for oil in a certain country will be changing at the rate of \(D^{\prime}(t)=(1+2 t)^{-1}\) billion barrels per year. Will more oil be consumed (demanded) during 2013 or during 2014 ? How much more?

FUTURE VALUE OF AN INCOME STREAM Money is transferred continuously into an account at the rate of \(5,000 e^{0.015 t}\) dollars per year at time \(t\) (years). The account earns interest at the annual rate of \(5 \%\) compounded continuously. How much will be in the account at the end of 3 years?

FUTURE VALUE OF AN INCOME STREAM Money is transferred continuously into an account at the constant rate of \(\$ 1,200\) per year. The account earns interest at the annual rate of \(8 \%\) compounded continuously. How much will be in the account at the end of 5 years?

PRESENT VALUE OF AN INCOME STREAM What is the present value of an investment that will generate income continuously at a constant rate of \(\$ 1,000\) per year for 10 years if the prevailing annual interest rate remains fixed at \(7 \%\) compounded continuously?

REAL ESTATE INVENTORY In a certain community the fraction of the homes placed on the market that remain unsold for at least \(t\) weeks is approximately \(f(t)=e^{-0.2 t}\). If 200 homes are currently on the market and if additional homes are placed on the market at the rate of 8 per week, approximately how many homes will be on the market 10 weeks from now?

AVERAGE REVENUE A bicycle manufacturer expects that \(x\) months from now consumers will be buying 5,000 bicycles per month at the price of \(P(x)=200+3 \sqrt{x}\) dollars per bicycle. What is the average revenue the manufacturer can expect from the sale of the bicycles over the next 16 months?

NUCLEAR WASTE A nuclear power plant produces radioactive waste at a constant rate of 300 pounds per year. The waste decays exponentially with a half-life of 35 years. How much of the radioactive waste from the plant will remain after 200 years?

GROWTH OF A TREE A tree has been transplanted and after \(x\) years is growing at the rate of $$ h^{\prime}(x)=0.5+\frac{1}{(x+1)^{2}} $$ meters per year. By how much does the tree grow during the second year?

FUTURE REVENUE A certain oil well that yields 900 barrels of crude oil per month will run dry in 3 years. The price of crude oil is currently \(\$ 92\) per barrel and is expected to rise at the constant rate of 80 cents per barrel per month. If the oil is sold as soon as it is extracted from the ground, how much total future revenue will be obtained from the well?

CONSUMERS' SURPLUS Suppose that the consumers' demand function for a certain commodity is \(D(q)=50-3 q-q^{2}\) dollars per unit. a. Find the number of units that will be bought if the market price is \(\$ 32\) per unit. b. Compute the consumer willingness to spend to get the number of units in part (a). c. Compute the consumers' surplus when the market price is \(\$ 32\) per unit. d. Use the graphing utility of your calculator to graph the demand curve. Interpret the consumer willingness to spend and the consumers' surplus as areas in relation to this curve.

AVERAGE PRICE Records indicate that \(t\) months after the beginning of the year, the price of bacon in local supermarkets was \(P(t)=0.06 t^{2}-0.2 t+6.2\) dollars per pound. What was the average price of bacon during the first 6 months of the year?

SURFACE AREA OF A HUMAN BODY The surface area \(S\) of the body of an average person 4 feet tall who weighs \(w \mathrm{lb}\) changes at the rate $$ S^{\prime}(w)=110 w^{-0.575} \quad \text { in }^{2} / \mathrm{lb} $$ The body of a particular child who is 4 feet tall and weighs \(50 \mathrm{lb}\) has surface area \(1,365 \mathrm{in}^{2}\). If the child gains \(3 \mathrm{lb}\) while remaining the same height, by how much will the surface area of the child's body increase?

TEMPERATURE CHANGE At \(t\) hours past midnight, the temperature \(T\left({ }^{\circ} \mathrm{C}\right)\) in a certain northern city is found to be changing at a rate given by \(T^{\prime}(t)=-0.02(t-7)(t-14) \quad{ }^{\circ}\) C/hour By how much does the temperature change between 8 A.M. and 8 P.M.?

EFFECT OF A TOXIN A toxin is introduced to a bacterial colony, and \(t\) hours later, the population \(P(t)\) of the colony is changing at the rate $$ \frac{d P}{d t}=-(\ln 3) 3^{4-t} $$ If there were 1 million bacteria in the colony when the toxin was introduced, what is \(P(t)\) ? [Hint: Note that \(3^{x}=e^{x \ln 3}\).]

INVESTING IN A DOWN MARKET PERIOD Jan opens a stock account with \(\$ 5,000\) at the beginning of January and subsequently deposits \(\$ 200\) a month. Unfortunately, the market is depressed, and she finds that \(t\) months after depositing a dollar, only \(100 f(t)\) cents remain, where \(f(t)=e^{-0.01 t}\). If this pattern continues, what will her account be worth after 2 years? [Hint: Think of this as a survival and renewal problem.]

DISTANCE AND VELOCITY After \(t\) minutes, an object moving along a line has velocity \(v(t)=1+4 t+3 t^{2}\) meters per minute. How far does the object travel during the third minute?

AVERAGE POPULATION The population (in thousands) of a certain city \(t\) years after January 1 , 2005 , is given by the function $$ P(t)=\frac{150 e^{0.03 t}}{1+e^{0.03 t}} $$ What is the average population of the city during the decade \(2005-2015 ?\)

DISTRIBUTION OF INCOME A study suggests that the distribution of incomes for social workers and physical therapists may be represented by the Lorenz curves \(y=L_{1}(x)\) and \(y=L_{2}(x)\), respectively, where $$ L_{1}(x)=x^{1.6} \text { and } L_{2}(x)=0.65 x^{2}+0.35 x $$ For which profession is the distribution of income more equitable?

DISTRIBUTION OF INCOME A study conducted by a certain state determines that the Lorenz curves for high school teachers and real estate brokers are given by the functions $$ \begin{aligned} &L_{1}(x)=0.67 x^{4}+0.33 x^{3} \\ &L_{2}(x)=0.72 x^{2}+0.28 x \end{aligned} $$ respectively. For which profession is the distribution of income more equitable?

CONSERVATION A lake has roughly the same shape as the bottom half of the solid formed by rotating the curve \(2 x^{2}+3 y^{2}=6\) about the \(x\) axis, for \(x\) and \(y\) measured in miles. Conservationists want the lake to contain 1,000 trout per cubic mile. If the lake currently contains 5,000 trout, how many more must be added to meet this requirement?

HORTICULTURE A sprinkler system sprays water onto a garden in such a way that \(11 e^{-r^{2} / 10}\) inches of water per hour are delivered at a distance of \(r\) feet from the sprinkler. What is the total amount of water laid down by the sprinkler within a 5 -foot radius during a 20 -minute watering period?

SPEED AND DISTANCE A car is driven so that after \(t\) hours its speed is \(S(t)\) miles per hour. a. Write down a definite integral that gives the average speed of the car during the first \(N\) hours. b. Write down a definite integral that gives the total distance the car travels during the first \(N\) hours. c. Discuss the relationship between the integrals in parts (a) and (b).

Use the graphing utility of your calculator to draw the graphs of the curves \(y=-x^{3}-2 x^{2}+5 x-2\) and \(y=x \ln x\) on the same screen. Use ZOOM and TRACE or some other feature of your calculator to find where the curves intersect, and then compute the area of the region bounded by the curves.