Q: A family wants to invest an amount now so that it grows to $10,000$ in $20$ years at a continuous annual interest rate of $4\%$. What initial investment should they make?

$10,000 \times e^{-0.04 \times 20}$

Q: A bank offers a continuous interest rate of $6\%$ per year. How much should you invest now to have $10,000$ in $20$ years?

$10,000 \times e^{-0.06 \times 20}$

Question 1

Which of the following differential equations models exponential decay?

Accepted Answer

$\frac{dP}{dt} = -kP$

Question 2

Given the exponential decay equation $P(t) = P_0 e^{-kt}$, what does $P_0$ represent?

Accepted Answer

The initial amount present at time $t = 0$

Question 3

If the half-life $T$ of a substance is 10 years, what is its decay rate $k$?

Accepted Answer

$k = \frac{\ln 2}{10}$

Question 4

A radioactive isotope has a decay rate of $0.023$ per year. What is its half-life?

Accepted Answer

$T = \frac{\ln 2}{0.023}$

Question 5

Which formula would you use to determine the present value $P_0$ of an amount $P$ due $t$ years later at a continuous interest rate $k$?

Accepted Answer

$P_0 = P e^{-kt}$

Question 6

A family wants to invest an amount now so that it grows to $10,000$ in $20$ years at a continuous annual interest rate of $4\%$. What initial investment should they make?

Accepted Answer

$10,000 	imes e^{-0.04 	imes 20}$

Question 7

The temperature $T$ of a cooling object in a room of temperature $C$ follows Newton’s Law of Cooling: $\frac{dT}{dt} = -k(T - C)$. What is the general solution for $T(t)$?

Accepted Answer

$T(t) = a e^{-kt} + C$

Question 8

A forensic scientist finds a body at noon with a temperature of $94.6^\circ$F. One hour later, the temperature is $93.4^\circ$F. The room temperature is $70^\circ$F. Using Newton’s Law of Cooling, what is the value of $a$ in the solution $T(t) = a e^{-kt} + 70$?

Accepted Answer

$a = 24.6$

Question 9

If the half-life of carbon-14 is $5730$ years, what is its decay rate $k$?

Accepted Answer

$k = \frac{\ln 2}{5730}$

Question 10

A sample has lost $60\%$ of its carbon-14. If the decay rate is $k = 0.00012097$, how old is the sample?

Accepted Answer

$t = \frac{\ln 0.4}{-0.00012097}$

Question 11

Which of the following best describes the relationship between decay rate $k$ and half-life $T$?

Accepted Answer

$T = \frac{\ln 2}{k}$

Question 12

If $P(t) = P_0 e^{-kt}$ and $P(t)$ is half of $P_0$ at time $t = T$, which equation must be satisfied?

Accepted Answer

$e^{-kT} = \frac{1}{2}$

Question 13

A scientist finds that a linen sample contains $77.7\%$ of its original carbon-14. If the decay rate is $k = 0.00012097$, how old is the sample?

Accepted Answer

$t = \frac{\ln 0.777}{-0.00012097}$

Question 14

Which scenario is best modeled by the equation $P(t) = P_0 e^{-kt}$?

Accepted Answer

Radioactive decay of a substance

Question 15

If the temperature of a body is $98.6^\circ$F at time of death, and $T(t) = 24.6 e^{-0.05 t} + 70$, how long ago did death occur if the body is found at $94.6^\circ$F?

Accepted Answer

$t = \frac{\ln \frac{24.6}{28.6}}{-0.05}$

Question 16

A bank offers a continuous interest rate of $6\%$ per year. How much should you invest now to have $10,000$ in $20$ years?

Accepted Answer

$10,000 	imes e^{-0.06 	imes 20}$

Question 17

Which of the following is NOT a correct application of exponential decay?

Accepted Answer

Predicting the linear decrease in water level in a tank

Exponential and Logarithmic Functions: Applications in Business Calculus

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