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 $$94.6^\circF. $$One hour later, the temperature is $$93.4^\circF. $$The room temperature is $$70^\circF. $$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 $$98.6^\circF at $$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 $$94.6^\circF$$?

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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