91Ó°ÊÓ

Let \(P(t)\) be the population (in millions) of a certain city \(t\) years after 1990, and suppose that \(P(t)\) satisfies the differential equation $$ P^{\prime}(t)=.02 P(t), \quad P(0)=3 $$ (a) Find the formula for \(P(t)\). (b) What was the initial population, that is, the population in \(1990 ?\) (c) What is the growth constant? (d) What was the population in \(1998 ?\) (e) Use the differential equation to determine how fast the population is growing when it reaches 4 million people. (f) How large is the population when it is growing at the rate of 70,000 people per year?

After \(t\) hours there are \(P(t)\) cells present in a culture, where \(P(t)=5000 e^{.2 t}\). (a) How many cells were present initially? (b) Give a differential equation satisfied by \(P(t)\). (c) When will the population double? (d) When will 20,000 cells be present?

Ten thousand dollars is deposited in a savings account at \(4.6 \%\) interest compounded continuously. (a) What differential equation is satisfied by \(A(t)\), the balance after \(t\) years? (b) What is the formula for \(A(t)\) ? (c) How much money will be in the account after 3 years? (d) When will the balance triple? (e) How fast is the balance growing when it triples?

An investment earns \(5.1 \%\) interest compounded continuously and is currently growing at the rate of $$\$ 765$$ per year. What is the current value of the investment?

The rate of growth of a certain cell culture is proportional to its size. In 10 hours a population of 1 million cells grew to 9 million. How large will the cell culture be after 15 hours?

How many years are required for an investment to double in value if it is appreciating at the rate of \(4 \%\) compounded continuously?

After a drug is taken orally, the amount of the drug in the bloodstream after \(t\) hours is \(f(t)=122\left(e^{-.2 t}-e^{-t}\right)\) units. (a) Graph \(f(t), f^{\prime}(t)\), and \(f^{\prime \prime}(t)\) in the window \([0,12]\) by \([-20,75] .\) (b) How many units of the drug are in the bloodstream after 7 hours? (c) At what rate is the level of drug in the bloodstream increasing after 1 hour? (d) While the level is decreasing, when is the level of drug in the bloodstream 20 units? (e) What is the greatest level of drug in the bloodstream, and when is this level reached? (f) When is the level of drug in the bloodstream decreasing the fastest?

For each demand function, find \(E(p)\) and determine if demand is elastic or inelastic (or neither) at the indicated price. $$ q=700-5 p, p=80 $$

If an investment triples in 15 years, what interest rate (compounded continuously) does the investment earn?

The decay constant for the radioactive element cesium 137 is .023 when time is measured in years. Find its half-life.