🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus for Business, Economics, Life Sciences, and Social Sciences 13th ## Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen ## Chapter 3 ## Additional Derivative Topics ## Educators  ### Problem 1 Solve for the variable to two decimal places. $$A=1,200 e^{0.04(5)}$$ Kaylee M. Numerade Educator ### Problem 2 Solve for the variable to two decimal places. $$A=3,000 e^{0.07(10)}$$ Kaylee M. Numerade Educator ### Problem 3 Solve for the variable to two decimal places. $$9827.30=P e^{0.025(3)}$$ Kaylee M. Numerade Educator ### Problem 4 Solve for the variable to two decimal places. $$50,000=P e^{0.054(7)}$$ Kaylee M. Numerade Educator ### Problem 5 Solve for the variable to two decimal places. $$6,000=5,000 e^{0.0325 t}$$ Kaylee M. Numerade Educator ### Problem 6 Solve for the variable to two decimal places. $$10,0000=7,500 e^{0.085 t}$$ Kaylee M. Numerade Educator ### Problem 7 Solve for the variable to two decimal places. $$956=900 e^{1.5 r}$$ Kaylee M. Numerade Educator ### Problem 8 Solve for the variable to two decimal places. $$4,840=3,750 e^{4.25 r}$$ Kaylee M. Numerade Educator ### Problem 9 Use a calculator to evaluate A to the nearest cent in Problems 9 and 10. $$A=\ 1,000 e^{0.1 t} \text { for } t=2,5, \text { and } 8$$ Kaylee M. Numerade Educator ### Problem 10 Use a calculator to evaluate A to the nearest cent in Problems 9 and 10. $$A=\ 5,000 e^{0.08 t} \text { for } t=1,4, \text { and } 10$$ Kaylee M. Numerade Educator ### Problem 11 If$\$6,000$ is invested at $10 \%$ compounded continuously, graph the amount in the account as a function of time for a period of 8 years. Kaylee M.

(B) How long will it take for the account to be worth $\$ 11.000 ?$ Kaylee M. Numerade Educator ### Problem 27 A note will pay$\$20,000$ at maturity 10 years from now. How much should you be willing to pay for the note now if money is worth $5.2 \%$ compounded continuously? Kaylee M.

A note will pay $\$ 50,000$at maturity 5 years from now. How much should you be willing to pay for the note now if money is worth$6.4 \%$compounded continuously? Kaylee M. Numerade Educator ### Problem 29 An investor bought stock for$\$20,000 .$ Five years later, the stock was sold for $\$ 30,000$. If interest is compounded continuously, what annual nominal rate of interest did the original$\$20,000$ investment earn? Kaylee M.

A family paid $\$ 99,000$cash for a house. Fifteen years later, the house was sold for$\$195,000 .$ If interest is compounded continuously, what annual nominal rate of interest did the original $\$ 99,000$investment earn? Kaylee M. Numerade Educator ### Problem 31 Solving$A=P e^{r l}$for$P,$we obtain $$P=A e^{-r t}$$ which is the present value of the amount$A$due in$t$years if money earns interest at an annual nominal rate$r$compounded continuously. (A) Graph$P=10,000 e^{-0.08 t}, 0 \leq t \leq 50 .$(B)$\lim _{t \rightarrow \infty} 10,000 e^{-0.08 t}=?$[Guess, using part (A). [Conclusion: The longer the time until the amount$A$is due, the smaller is its present value, as we would expect. Kaylee M. Numerade Educator ### Problem 32 Referring to Problem 31 , in how many years will the$\$10,000$ be due in order for its present value to be $\$ 5.000 ?$ Kaylee M. Numerade Educator ### Problem 33 How long will it take money to double if it is invested at$4 \%$compounded continuously? Kaylee M. Numerade Educator ### Problem 34 How long will it take money to double if it is invested at$5 \%$compounded continuously? Kaylee M. Numerade Educator ### Problem 35 At what nominal rate compounded continuously must money be invested to double in 8 years? Kaylee M. Numerade Educator ### Problem 36 At what nominal rate compounded continuously must money be invested to double in 10 years? Kaylee M. Numerade Educator ### Problem 37 A man with$\$20,000$ to invest decides to diversify his investments by placing $\$ 10,000$in an account that earns$7.2 \%$compounded continuously and$\$10,000$ in an account that earns $8.4 \%$ compounded annually. Use graphical approximation methods to determine how long it will take for his total investment in the two accounts to grow to $\$ 35.000$ Kaylee M. Numerade Educator ### Problem 38 A woman invests$\$5,000$ in an account that earns $8.8 \%$ compounded continuously and $\$ 7,000$in an account that earns$9.6 \%$compounded annually. Use graphical approximation methods to determine how long it will take for her total investment in the two accounts to grow to$\$20,000$. Kaylee M.

### Problem 39

(A) Show that the doubling time $t$ (in years) at an annual rate $r$ compounded continuously is given by$$t=\frac{\ln 2}{r}$$
(B) Graph the doubling-time equation from part (A) for $0.02 \leq r \leq 0.30 .$ Is this restriction on $r$ reasonable? Explain.
(C) Determine the doubling times (in years, to two decimal places) for $r=5 \%, 10 \%, 15 \%, 20 \%, 25 \%,$ and $30 \%$. Kaylee M.

### Problem 40

Doubling rates
(A) Show that the rate $r$ that doubles an investment at continuously compounded interest in $t$ years is given by
$$r=\frac{\ln 2}{t}$$
(B) Graph the doubling-rate equation from part (A) for $1 \leq t \leq 20$. Is this restriction on $t$ reasonable? Explain.
(C) Determine the doubling rates for $t=2,4,6,8,10,$ and 12 years. Kaylee M.

### Problem 41

A mathematical model for the decay of radioactive substances is given by
$$Q=Q_{0} e^{r t}$$
where
\begin{aligned} Q_{0} &=\text { amount of the substance at time } t=0 \\ r &=\text { continuous compound rate of decay } \\ t &=\text { time in years } \\ Q &=\text { amount of the substance at time } t \end{aligned}
If the continuous compound rate of decay of radium per year is $r=-0.0004332$, how long will it take a certain amount of radium to decay to half the original amount? (This period is the half-life of the substance.) Kaylee M.

### Problem 42

The continuous compound rate of decay of carbon- 14 per year is $r=-0.0001238$. How long will it take a certain amount of carbon- 14 to decay to half the original amount? (Use the radioactive decay model in Problem $41 .$ Kaylee M.

### Problem 43

A cesium isotope has a half-life of 30 years. What is the continuous compound rate of decay? (Use the radioactive decay model in Problem $41 .$ Kaylee M.

### Problem 44

A strontium isotope has a half-life of 90 years. What is the continuous compound rate of decay? (Use the radioactive decay model in Problem $41 .)$ Kaylee M.

### Problem 45

A mathematical model for world population growth over short intervals is given by
$$P=P_{0} e^{r t}$$
where
\begin{aligned}P_{0} &=\text { population at time } t=0 \\r &=\text { continuous compound rate of growth} \\t &=\text { time in years } \\P &=\text { population at time } t\end{aligned}
How long will it take world population to double if it continues to grow at its current continuous compound rate of $1.3 \%$ per year? Kaylee M.

### Problem 46

How long will it take for the U.S. population to double if it continues to grow at a rate of $0.975 \%$ per year? Kaylee M.

### Problem 47

Some underdeveloped nations have population doubling times of 50 years. At what continuous compound rate is the population growing? (Use the population growth model in Problem $45 .$ ) Kaylee M.
Some developed nations have population doubling times of 200 years. At what continuous compound rate is the population growing? (Use the population growth model in Problem $45 .$ 