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 $$94.6^\circF. An $$hour later, the temperature is $$93.4^\circF. If $$the room temperature is $$70^\circF$$, 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 $$98.6^\circF$$, 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 $$70^\circF $$and the body temperature drops from $$98.6^\circF to$$ $$94.6^\circF in 3 $$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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