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 $t=0$

Question 3

If a radioactive substance has a decay rate $k = 0.023$ per year, what is its half-life?

Accepted Answer

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

Question 4

A sample of cesium-137 has a half-life of approximately 30.1 years. What is its decay rate $k$?

Accepted Answer

$k = \frac{\ln 2}{30.1}$

Question 5

A geologist finds that a linen wrapping has lost 22.3% of its carbon-14. If the half-life of carbon-14 is 5730 years, what is the decay rate $k$?

Accepted Answer

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

Question 6

If $40\%$ of the original carbon-14 remains in a skeleton, and the decay rate is $k = 0.00012097$ per year, how old is the skeleton?

Accepted Answer

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

Question 7

A parent wants to invest an amount $P_0$ so that it grows to $10,000$ in 20 years at a continuously compounded annual rate of $4\%$. What equation should be used to solve for $P_0$?

Accepted Answer

$10,000 = P_0 e^{0.04 \cdot 20}$

Question 8

If you want $10,000$ in 20 years and the interest rate is $6\%$ compounded continuously, what initial investment $P_0$ is required?

Accepted Answer

$P_0 = 10,000 e^{-0.06 \cdot 20}$

Question 9

Newton’s Law of Cooling states that the rate of change of temperature $T$ of an object is proportional to:

Accepted Answer

The difference between the object’s temperature and the surroundings

Question 10

A body is found at noon with a temperature of $94.6^\circ$F. An hour later, the temperature is $93.4^\circ$F. If the room temperature is $70^\circ$F, what is the value of $a$ in the solution $T(t) = a e^{-kt} + 70$?

Accepted Answer

$a = 24.6$

Question 11

Given $T(t) = 24.6 e^{-kt} + 70$ and $T(1) = 93.4$, what equation can be used to solve for $k$?

Accepted Answer

$93.4 = 24.6 e^{-k} + 70$

Question 12

If the normal body temperature is $98.6^\circ$F, and $T(t) = 24.6 e^{-0.05t} + 70$, how do you solve for the time $t$ since death?

Accepted Answer

$98.6 = 24.6 e^{-0.05t} + 70$

Question 13

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

Accepted Answer

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

Question 14

A radioactive isotope decays according to $P(t) = P_0 e^{-kt}$. If $P(10) = \frac{1}{4} P_0$, what is $k$?

Accepted Answer

$k = \frac{\ln 4}{10}$

Question 15

If a substance decays at a rate of $5.3\%$ per day, what is its half-life in days?

Accepted Answer

$T = \frac{\ln 2}{0.053}$

Question 16

A $10,000$ payment is due in 20 years. If the interest rate is $4\%$ compounded continuously, what is the present value?

Accepted Answer

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

Question 17

Which of the following is the correct solution to the differential equation $\frac{dP}{dt} = -kP$ with initial condition $P(0) = P_0$?

Accepted Answer

$P(t) = P_0 e^{-kt}$

Question 18

A forensic scientist uses Newton’s Law of Cooling to estimate time of death. If the room temperature is $70^\circ$F and the body temperature drops from $98.6^\circ$F to $94.6^\circ$F in 3 hours, what is the value of $a$ in $T(t) = a e^{-kt} + 70$?

Accepted Answer

$a = 28.6$

Question 19

If $N = N_0 e^{-kt}$ and $N/N_0 = 0.777$, $k = 0.00012097$, what is the age $t$ of the sample?

Accepted Answer

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

Exponential and Logarithmic Functions: Applications in Business Calculus

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