91Ó°ÊÓ

Investment Growth A large corporation starts at time \(t=0\) to invest part of its receipts continuously at a rate of \(P\) dollars per year in a fund for future corporate expansion. Assume that the fund earns \(r\) percent interest per year compounded continuously. So, the rate of growth of the amount \(A\) in the fund is given by \(d A / d t=r A+P,\) where \(A=0\) when \(t=0 .\) Solve this differential equation for \(A\) as a function of \(t .\)

A 200 -gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time \(t=0\) , distilled water is admitted to the tank at a rate of 10 gallons per minute, and the well-stirred solution is withdrawn at the same rate. (a) Find the amount of concentrate \(Q\) (in pounds) in the solution as a function of \(t\) . (b) Find the time at which the amount of concentrate in the tank reaches 15 pounds. (c) Find the amount of concentrate (in pounds) in the solution as \(t \rightarrow \infty .\)

A 200 -gallon tank is half full of distilled water. Starting at time \(t=0,\) a solution containing 0.5 pound of concentrate per gallon is admitted to the tank at a rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at a rate of 3 gallons per minute. (a) At what time will the tank be full? (b) At the time the tank is full, how many pounds of concentrate will it contain? (c) Repeat parts (a) and (b), assuming that the solution entering the tank contains 1 pound of concentrate per gallon.

Chemical Reaction In a chemical reaction, a certain compound changes into another compound at a rate proportional to the unchanged amount. There is 40 grams of the original compound initially and 35 grams after 1 hour. When will 75 percent of the compound be changed?

Compound Interest In Exercises \(45-48,\) find the principal \(P\) that must be invested at rate \(r\) , compounded monthly, so that \(\$ 1,000,000\) will be available for retirement in \(t\) years. $$r=8 \%, \quad t=35$$

Finding Orthogonal Trajectories In Exercises 43-48, find the orthogonal trajectories for the family of curves. Use a graphing utility to graph several members of each family. $$y=C e^{x}$$

In Exercises \(47-54,\) solve the Bernoulli differential equation. \(y^{\prime}+\left(\frac{1}{x}\right) y=x y^{2}\)

Population In Exercises \(51-54,\) the population (in millions) of a country in 2015 and the expected continuous annual rate of change \(k\) of the population are given. (Source: U.S. Census Bureau, International Data Base) (a) Find the exponential growth model \(P=C e^{k t}\) for the population by letting \(t=5\) correspond to \(2015 .\) (b) Use the model to predict the population of the country in \(2030 .\) (c) Discuss the relationship between the sign of \(k\) and the change in population for the country. $$\begin{array}{l}{\text { Country }}&{\text { 2015 Population }}&&{\text { \(k\) }} \\ {Latvia}& {\text {2.0}}&&{{-0.011}}\end{array}

Using a Logistic Equation In Exercises 53 and 54 , the logistic equation models the growth of a population. Use the equation to (a) find the value of \(k,(\) b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach 50% of its carrying capacity, and (e) write a logistic differential equation that has the solution \(P(t).\) $$P(t)=\frac{2100}{1+29 e^{-0.75 t}}$$

Using a Logistic Equation In Exercises 53 and 54 , the logistic equation models the growth of a population. Use the equation to (a) find the value of \(k,(\) b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach 50% of its carrying capacity, and (e) write a logistic differential equation that has the solution \(P(t).\) $$P(t)=\frac{5000}{1+39 e^{-0.2 t}}$$