91Ó°ÊÓ

Savings Account Let \(A(t)\) be the balance in a savings account after \(t\) years, and suppose that \(A(t)\) satisfies the differential equation $$ A^{\prime}(t)=.045 A(t), \quad A(0)=3000 $$ (a) How much money was originally deposited in the account? (b) What interest rate is being earned? (c) Find the formula for \(A(t)\) (d) What is the balance after 5 years? (e) Use part (d) and the differential equation to determine how fast the balance is growing after 5 years. (f) How large will the balance be when it is growing at the rate of \(\$ 270\) per year?

Ten thousand dollars is deposited in a savings account at \(4.6 \%\) yearly 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?

Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated. $$g(p)=5 /(2 p+3) \text { at } p=1 \text { and } p=11$$

Ten thousand dollars is invested at \(6.5 \%\) interest compounded continuously. When will the investment be worth \(\$ 41,787 ?\)

Solve the given differential equation with initial condition. $$y^{\prime}=3 y, y(0)=1$$

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

Growth Let \(P(t)\) be the population (in millions) of a certain city \(t\) years after \(2015,\) and suppose that \(P(t)\) satisfies the differential equation $$P^{\prime}(t)=.01 P(t), P(0)=2$$ (a) Find a formula for \(P(t).\) (b) What was the initial population, that is, the population in 2015 ? (c) Estimate the population in 2019.

A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let \(P(t)\) denote the number of fruit flies \(t\) days later, and let \(k=.08\) denote the growth constant. (a) Write a differential equation and initial condition that model the growth of this colony. (b) Find a formula for \(P(t).\) (c) Estimate the size of the colony 5 days later.

The initial size of a bacteria culture that grows exponentially was 10,000. After 1 day, there are 15,000 bacteria. (a) Find the growth constant if time is measured in days. (b) How long will it take for the culture to double in size?

After \(t\) hours there are \(P(t)\) cells present in a culture, where \(P(t)=5000 e^{0.2 t}\) (a) How many cells were present initially? (b) Give a differential equation satisfied by \(P(t)\) (c) When will the initial number of cells double? (d) When will 20,000 cells be present?