Q: Jean-Sylvain and Annick want to invest an amount $ P_0 $ at 4% annual interest, compounded continuously, so that it grows to $ 10,000 $ in 20 years. What should $ P_0 $ be?

$ P_0 = 10,000 e^{-0.04 \times 20} \approx 4493.29 $

Q: If the interest rate is increased to 6% per year, compounded continuously, what initial investment is needed to reach $ 10,000 $ in 20 years?

$ P_0 = 10,000 e^{-0.06 \times 20} \approx 3011.94 $

Question 1

Which of the following differential equations models exponential decay?

Accepted Answer

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

Question 2

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

Accepted Answer

The initial amount present at $ t = 0 $

Question 3

If the half-life $ T $ of a substance is known, which formula gives the decay rate $ k $?

Accepted Answer

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

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} \approx 30.1 $ years

Question 5

The half-life of a certain radioactive element is 13 days. What is its decay rate $ k $ (per day)?

Accepted Answer

$ k = \frac{\ln 2}{13} \approx 0.053 $

Question 6

A sample of carbon-14 has a half-life of 5730 years. What is the decay rate $ k $ (per year)?

Accepted Answer

$ k = \frac{\ln 2}{5730} \approx 0.00012097 $

Question 7

Archaeologists find that a linen wrapping has lost 22.3% of its carbon-14. If the half-life of carbon-14 is 5730 years, how old is the linen? (Round to the nearest year)

Accepted Answer

$ t = \frac{\ln 0.777}{-0.00012097} \approx 2086 $ years

Question 8

A skeleton is found to have lost 60% of its carbon-14. Using a half-life of 5730 years, approximately how old is the skeleton?

Accepted Answer

$ t = \frac{\ln 0.4}{-0.00012097} \approx 7574 $ years

Question 9

Jean-Sylvain and Annick want to invest an amount $ P_0 $ at 4% annual interest, compounded continuously, so that it grows to $ 10,000 $ in 20 years. What should $ P_0 $ be?

Accepted Answer

$ P_0 = 10,000 e^{-0.04 	imes 20} \approx 4493.29 $

Question 10

If the interest rate is increased to 6% per year, compounded continuously, what initial investment is needed to reach $ 10,000 $ in 20 years?

Accepted Answer

$ P_0 = 10,000 e^{-0.06 	imes 20} \approx 3011.94 $

Question 11

Which formula gives the present value $ P_0 $ of an amount $ P $ due in $ t $ years at a continuous interest rate $ k $?

Accepted Answer

$ P_0 = P e^{-kt} $

Question 12

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 13

A body is found at noon with a temperature of $ 94.6^\circ $F. An 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 14

Given $ T(0) = 94.6 $ and $ T(1) = 93.4 $ for a cooling body in a $ 70^\circ $F room, what is the value of $ k $ in $ T(t) = 24.6 e^{kt} + 70 $? (Round to two decimal places)

Accepted Answer

$ k = \ln(23.4/24.6) \approx -0.05 $

Question 15

If the normal body temperature is $ 98.6^\circ $F, and the model $ T(t) = 24.6 e^{-0.05t} + 70 $ is used, how many hours before noon did death occur? (Round to the nearest hour)

Accepted Answer

$ t = \frac{\ln(28.6/24.6)}{0.05} \approx 3 $ hours

Question 16

Which of the following best describes the solution to the differential equation $ \frac{dP}{dt} = -kP $?

Accepted Answer

An exponential decay function

Question 17

A certain medication in the bloodstream decays exponentially with a half-life of 8 hours. If the initial dose is 200 mg, how much remains after 24 hours?

Accepted Answer

$ 200 	imes e^{-k 	imes 24} = 200 	imes (1/2)^{24/8} = 200 	imes (1/2)^3 = 25 $ mg

Exponential and Logarithmic Functions: Applications of Exponential Decay

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