Kaylee M.

Kaylee M.

Kaylee M.

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 $45 .$