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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