Question 1

Which of the following correctly rewrites $$Q = a b^t in $$the form $$Q = a e$$^{kt}$$?

Accepted Answer

$$Q = a e$$^{t \ln b}$$

Question 2

Given $$Q = 120(0.97)^t$$, convert this to the form $$Q = a e$$^{kt}$$. What is the value of k$?

Accepted Answer

$$k = \ln(0.97)$$

Question 3

A population of bacteria is modeled by $$Q = 160 e$$^{0.087t}$$. What is the initial population?

Accepted Answer

$$160$$

Question 4

If $$Q = 290(1.827)^t$$, what is the continuous growth rate $$k in $$the equivalent form $$Q = a e$$^{kt}$$?

Accepted Answer

$$k = \ln(1.827)$$

Question 5

A medication decays in the bloodstream according to $$Q = 120(0.97)^t. $$After 5 hours, what percentage of the medication remains?

Accepted Answer

$$0.97^5$$ 	imes 100\%$$

Question 6

If a population grows according to $$Q = a e$$^{kt}$$ and doubles every 3 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{3}$$

Question 7

Given $$Q = 160 e$$^{0.087t}$$, how long will it take for the quantity to triple?

Accepted Answer

$$t = \frac{\ln 3}{0.087}$$

Question 8

A radioactive substance decays according to $$Q = a e$$^{kt}$$, where k$$ \u003c 0$$. If the half-life is 10 years, what is k$?

Accepted Answer

$$k = \frac{\ln(1/2)}{10}$$

Question 9

If $$Q = 290(1.827)^t$$, what is the value of $$Q$$ when $$t = 2$$?

Accepted Answer

$$290 	imes (1.827)^2$

Question 10

Which of the following best describes the relationship between the discrete growth rate $$b$$ and the continuous growth rate $$k in $$exponential models?

Accepted Answer

$$k = \ln b$$

Question 11

A scientist models the spread of a virus with $$Q = a e$$^{kt}$$. If the number of cases increases by 5% per day, what is the value of k$?

Accepted Answer

$$k = \ln(1.05)$$

Question 12

If $$Q = a b^t$$ and $$b \u003c 1$$, what does this indicate about the process being modeled?

Accepted Answer

The process is experiencing exponential decay.

Question 13

A population starts at 500 and grows according to $$Q = 500 e$$^{0.03t}$$. How long will it take for the population to reach 1000?

Accepted Answer

$$t = \frac{\ln(2)}{0.03}$$

Question 14

Which of the following is the correct conversion from $$Q = a e$$^{kt}$$ to Q = a b^t$?

Accepted Answer

$$b = e^k$$

Question 15

A chemical reaction decreases the concentration of a substance by 3% per minute. If the initial concentration is 200 mg/L, what is the concentration after 10 minutes?

Accepted Answer

$$200 	imes (0.97$$)^{10}$$

Question 16

If $$Q = a e$$^{kt}$$ and k = 0$, what happens to Q$ over time?

Accepted Answer

$$Q$$ remains constant at $$a$$

Question 17

A population is modeled by $$Q = 120(0.97)^t. $$What is the percentage decrease per time unit?

Accepted Answer

3%

Question 18

If $$Q = a b^t$$ and $$b = 1$$, what does this indicate about the modeled process?

Accepted Answer

The process is constant over time.

Question 19

A population of fish in a lake is modeled by $$Q = 500(1.05)^t. $$What is the population after 4 years?

Accepted Answer

$$500 	imes (1.05)^4$

Exponential Functions: Converting Forms and Growth Rates 5.2

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