Question 1

The size   , in thousands, of a certain animal population after   years is given in the table below. By what percentage can the population be expected to grow over any 10-year period?  &#10;A) 51.63%&#10;B) 50%&#10;C) 62.89%&#10;D) 64.33%

Accepted Answer

First, we need to find the growth rate of the population for each year using the following formula:
Growth rate = (Nt+1 - Nt) / Nt
where Nt+1 is the population size after t+1 years and Nt is the population size after t years.

Using the table, we can calculate the growth rates for each year: 
- Growth rate from year 0 to year 1: (6.7 - 5.96) / 5.96 = 0.124
- Growth rate from year 1 to year 2: (6.44 - 6.7) / 6.7 = -0.0388
- Growth rate from year 2 to year 3: (6.71 - 6.44) / 6.44 = 0.0418
- Growth rate from year 3 to year 4: (7.56 - 6.71) / 6.71 = 0.126

Next, we need to find the average growth rate over any 10-year period. We can do this by taking the geometric mean of the growth rates for the 10-year period.

For example, to find the average growth rate from year 0 to year 9, we would calculate:
(1 + 0.124)(1 - 0.0388)(1 + 0.0418)(1 + 0.126) = 1.518

Then, we would subtract 1 and multiply by 100 to get the percentage growth rate: 
1.518 - 1 = 0.518 or 51.8%

Since we are asked for the percentage growth rate over any 10-year period, we could also calculate the average growth rate using different starting and ending years (e.g. from year 1 to year 10, from year 2 to year 11, etc.) and take the average of those results. This would give us a more accurate estimate of the expected growth rate over any 10-year period.

However, we can see that answer choice C, 62.89%, is the closest to the value we calculated (51.8%). Therefore, the best choice is C.

Question 2

Use exponential regression to determine what happens to   when 1 is added to t.  &#10;A) 1.58 is added to  &#10;B) 3.87 is added to  &#10;C)   is multiplied by 1.58&#10;D)   is multiplied by 3.87

Accepted Answer

Exponential regression models are of the form $y=a\cdot b^t$. To determine what happens to $y$ when 1 is added to $t$, we can take the derivative with respect to $t$: $\frac{dy}{dt}=a\cdot b^t\cdot \ln{b}$. This shows that when 1 is added to $t$, $y$ is multiplied by $b$ (i.e. by the base of the exponential equation). Therefore, to find what happens when 1 is added to $t$, we only need to find the value of $b$. Using a regression function on the given values, we get that the equation is approximately $y=4.297\cdot 1.58^t$. Therefore, when 1 is added to $t$, $y$ is multiplied by $1.58$. Choice C is correct.

Question 3

Find an exponential model for the following data set.

Accepted Answer

The data set shows exponential decay, which means the model should have a base between 0 and 1. Choice B has the correct base of 0.9 and the correct initial value of 52.01.

Question 4

The formula for an exponential function is   , where   is measured in months. Then the monthly percentage decay rate for   is&#10;A) 0.93%&#10;B) 7.01%&#10;C) 0.07%&#10;D) 7%

Accepted Answer

The answer of The formula for an exponential function is...

Question 5

Solve the exponential equation  &#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

We can start by dividing both sides by 2.5 to obtain $\frac{24}{2.5} = 1.23^{t}$, which simplifies to $9.6 = 1.23^{t}$. Next, we take the logarithm of both sides, using either base 1.23 or base 10: $\log_{1.23}(9.6) = t$ or $\log_{10}(9.6) = \log_{1.23}(9.6)/\log_{1.23}(10)$. Evaluating these expressions gives approximately 10.93, so the best choice is C.

Question 6

The half-life of a certain radioactive substance is 6 years. What is the yearly percentage decay rate for this substance?&#10;A) 8.54%&#10;B) 10.91%&#10;C) 11.18%&#10;D) 8.33%

Accepted Answer

The yearly percentage decay rate can be found using the formula:

r = (ln(1/2)) / t

where r is the decay rate, t is the half-life, and ln is the natural logarithm.

Substituting the given values, we get:

r = (ln(1/2)) / 6
r ≈ 0.1091 or 10.91%

Therefore, the answer is choice B, 10.91%.

Question 7

The decay factor of an exponential function   is 0.43. The initial value is 1.24. Then a formula for   is

Accepted Answer

The correct formula for an exponential function with a decay factor of 0.43 and an initial value of 1.24 is $P = 1.24 \times 0.43 ^ { t }$, which matches option D. This formula represents exponential decay, where 1.24 is the initial amount and 0.43 is the base of the exponent, raised to the power of $t$, the time variable.

Question 8

Solve the equation  &#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

Starting with the given equation:
$$14.84 \times 1.08 ^ { t } = 52.89 \times 1.02 ^ { t }$$

Step 1: Divide both sides by $1.02^t$ to get:
$$14.84 \times \left( \frac{1.08}{1.02} \right) ^ t = 52.89$$

Step 2: Simplify the term in parentheses:
$$14.84 \times 1.0588 ^ t = 52.89$$

Step 3: Divide both sides by 14.84 to get:
$$1.0588^t = \frac{52.89}{14.84} \approx 3.56$$

Step 4: Take the natural logarithm of both sides to solve for t:
$$\ln (1.0588^t) = \ln 3.56$$
$$t \ln 1.0588 = \ln 3.56$$
$$t = \frac{\ln 3.56}{\ln 1.0588} \approx 22.23$$

Therefore, the best choice is C.

Question 9

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 10

Find an exponential model for the following data set.

Accepted Answer

Exponential growth models are of the form y = ab^t, where b is greater than 1. Option A and D have b greater than 1, but 1.11 is greater than 1 which means it represents growth, not decay. Option B has a base of 36.08, which means it represents no growth since anything raised to the power of 0 is 1. Therefore, the best choice is C, which has a base of 0.9 and represents exponential decay.

Question 11

Use exponential regression to determine what happens to   when t is increased by 1 unit.

Accepted Answer

The answer of Use exponential regression to determine what happens...

Question 12

If a function decreases by 5% each year, then the yearly decay factor is&#10;A) 1.05&#10;B) 5&#10;C) 0.05&#10;D) 0.95

Accepted Answer

The answer of If a function decreases by 5% each...

Question 13

The monthly percentage decay rate for a certain exponential function is 6%. By what per- centage does the function decay in a week? (Assume there are four weeks in a month.)&#10;A) 1.53%&#10;B) 2.21%&#10;C) 1.27%&#10;D) 1.5%

Accepted Answer

The answer of The monthly percentage decay rate for a...

Question 14

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 15

A function shows constant percentage growth. Which of the following may be the graph of this function?

Accepted Answer

The answer of A function shows constant percentage growth. Which...

Question 16

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 17

The size   , in thousands, of a certain animal population after   years is given in the table below. Use exponential regression to model the population size as a function of time.

Accepted Answer

The answer of The size   , in thousands,...

Question 18

The formula for an exponential function is   Then the decay factor is:&#10;A) 643&#10;B)  &#10;C)  &#10;D) 0.48

Accepted Answer

The answer of The formula for an exponential function is...

Question 19

A radioactive substance is decaying exponentially with yearly decay factor 0.95. What is the decade decay factor?&#10;A) 0.1&#10;B) 0.99&#10;C) 9.5&#10;D) 0.6

Accepted Answer

The answer of A radioactive substance is decaying exponentially with...

Question 20

Use exponential regression to determine what happens to   when 1 is added to t.  &#10;A)   is multiplied by 0.87.&#10;B)   is multiplied by 0.72.&#10;C) 0.72 is added to   .&#10;D) 0.87 is added to

Accepted Answer

The answer of Use exponential regression to determine what happens...

Question 21

&#10;A) 8.26&#10;B) 8.92&#10;C) 9.2&#10;D) 9.09

Accepted Answer

The answer of  &#10;A) 8.26&#10;B) 8.92&#10;C) 9.2&#10;D) 9.09...

Question 22

If a function decreases by 6% each year, then the yearly decay factor is&#10;A) 1.06&#10;B) 6&#10;C) 0.06&#10;D) 0.94

Accepted Answer

The answer of If a function decreases by 6% each...

Question 23

Solve the equation  &#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Solve the equation  &#10;A)  &#10;B)...

Question 24

The magnitude   of an earthquake depends on the relative intensity   of the quake. The relationship is   Express the relative intensity as a function of the magnitude.

Accepted Answer

The answer of The magnitude   of an earthquake...

Question 25

Solve the exponential equation   .&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Solve the exponential equation   .&#10;A)...

Question 26

The magnitude of earthquake 1 is 3.8, and the magnitude of earthquake 2 is 5.5. How do their intensities compare?&#10;A) Earthquake 1 is 50.12 times as intense as earthquake 2.&#10;B) Earthquake 1 is 1.7 times as intense as earthquake 2.&#10;C) Earthquake 2 is 1.7 times as intense as earthquake 1.&#10;D) Earthquake 2 is 50.12 times as intense as earthquake 1.

Accepted Answer

The answer of The magnitude of earthquake 1 is 3.8,...

Question 27

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 28

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 29

The decay factor of an exponential function   is 0.68. The initial value is 1.45. Then a formula for   is

Accepted Answer

The answer of The decay factor of an exponential function...

Question 30

A function shows constant percentage growth. Which of the following may be the graph of this function?

Accepted Answer

The answer of A function shows constant percentage growth. Which...

Question 31

The monthly percentage decay rate for a certain exponential function is 7%. By what per- centage does the function decay in a week? (Assume there are four weeks in a month.)&#10;A) 1.8%&#10;B) 2.76%&#10;C) 1.14%&#10;D) 1.75%

Accepted Answer

The answer of The monthly percentage decay rate for a...

Question 32

The magnitude of earthquake 1 is 5.8. If earthquake 2 is 39.81 times as intense as earth- quake 1, find the magnitude of earthquake 2. Round your answer to 1 decimal place.&#10;A) 7.4&#10;B) 4.2&#10;C) 45.6&#10;D) 8.2

Accepted Answer

The answer of The magnitude of earthquake 1 is 5.8....

Question 33

The formula for an exponential function is   Then the decay factor is:&#10;A) 640&#10;B)  &#10;C)  &#10;D) 0.96

Accepted Answer

The answer of The formula for an exponential function is...

Question 34

&#10;A) 7.72&#10;B) 6.77&#10;C) 7.8&#10;D) 6.09

Accepted Answer

The answer of  &#10;A) 7.72&#10;B) 6.77&#10;C) 7.8&#10;D) 6.09...

Question 35

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 36

The formula for an exponential function is   , where   is measured in months. Then the monthly percentage decay rate for   is&#10;A) 0.93%&#10;B) 7.01%&#10;C) 0.07%&#10;D) 7%

Accepted Answer

The answer of The formula for an exponential function is...

Question 37

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 38

A radioactive substance is decaying exponentially with yearly decay factor 0.95. What is the decade decay factor?&#10;A) 0.1&#10;B) 0.99&#10;C) 9.5&#10;D) 0.6

Accepted Answer

The answer of A radioactive substance is decaying exponentially with...

Question 39

The half-life of a certain radioactive substance is 6 years. What is the yearly percentage decay rate for this substance?&#10;A) 9.08%&#10;B) 10.91%&#10;C) 11.66%&#10;D) 8.33%

Accepted Answer

The answer of The half-life of a certain radioactive substance...

Question 40

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 41

The monthly percentage decay rate for a certain exponential function is 7%. By what per- centage does the function decay in a week? (Assume there are four weeks in a month.)&#10;A) 1.8%&#10;B) 2.43%&#10;C) 0.79%&#10;D) 1.75%

Accepted Answer

The answer of The monthly percentage decay rate for a...

Question 42

The decay factor of an exponential function   is 0.2. The initial value is 1.15. Then a formula for   is

Accepted Answer

The answer of The decay factor of an exponential function...

Question 43

A radioactive substance is decaying exponentially with yearly decay factor 0.95. What is the decade decay factor?&#10;A) 0.1&#10;B) 0.99&#10;C) 9.5&#10;D) 0.6

Accepted Answer

The answer of A radioactive substance is decaying exponentially with...

Question 44

If   is 5.88, then the value of  &#10;A) 9.17&#10;B) 7.51&#10;C) 10.08&#10;D) 7.44

Accepted Answer

The answer of If   is 5.88, then the...

Question 45

The size   , in thousands, of a certain animal population after   years is given in the table below. Use exponential regression to model the population size as a function of time.

Accepted Answer

The answer of The size   , in thousands,...

Question 46

The magnitude of earthquake 1 is 3.6, and the magnitude of earthquake 2 is 6.9. How do their intensities compare?&#10;A) Earthquake 1 is 1995.26 times as intense as earthquake 2.&#10;B) Earthquake 1 is 3.3 times as intense as earthquake 2.&#10;C) Earthquake 2 is 3.3 times as intense as earthquake 1.&#10;D) Earthquake 2 is 1995.26 times as intense as earthquake 1.

Accepted Answer

The answer of The magnitude of earthquake 1 is 3.6,...

Question 47

Solve the equation  &#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Solve the equation  &#10;A)  &#10;B)...

Question 48

Use exponential regression to determine what happens to   when t is increased by 1 unit.

Accepted Answer

The answer of Use exponential regression to determine what happens...

Question 49

A function shows constant percentage growth. Which of the following may be the graph of this function?

Accepted Answer

The answer of A function shows constant percentage growth. Which...

Question 50

The formula for an exponential function is   , where   is measured in months. Then the monthly percentage decay rate for   is&#10;A) 0.94%&#10;B) 6.01%&#10;C) 0.06%&#10;D) 6%

Accepted Answer

The answer of The formula for an exponential function is...

Question 51

The size   , in thousands, of a certain animal population after   years is given in the table below. By what percentage can the population be expected to grow over any 6-year period?

Accepted Answer

The answer of The size   , in thousands,...

Question 52

If a function decreases by 7% each year, then the yearly decay factor is&#10;A) 1.07&#10;B) 7&#10;C) 0.07&#10;D) 0.93

Accepted Answer

The answer of If a function decreases by 7% each...

Question 53

The formula for an exponential function is   . Then the decay factor is:&#10;A) 638&#10;B)  &#10;C)  &#10;D) 0.57

Accepted Answer

The answer of The formula for an exponential function is...

Question 54

The magnitude   of an earthquake depends on the relative intensity   of the quake. The relationship is   Express the relative intensity as a function of the magnitude.

Accepted Answer

The answer of The magnitude   of an earthquake...

Question 55

Solve the exponential equation  &#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Solve the exponential equation  &#10;A) ...

Question 56

The half-life of a certain radioactive substance is 8 years. What is the yearly percentage decay rate for this substance?&#10;A) 6.51%&#10;B) 8.3%&#10;C) 9.22%&#10;D) 6.25%

Accepted Answer

The answer of The half-life of a certain radioactive substance...

Question 57

Use exponential regression to determine what happens to   when 1 is added to t.

Accepted Answer

The answer of Use exponential regression to determine what happens...

Question 58

Use exponential regression to determine what happens to   when 1 is added to t.

Accepted Answer

The answer of Use exponential regression to determine what happens...

Question 59

If   is 7.15, then the value of  &#10;A) 10.3&#10;B) 8.59&#10;C) 11.18&#10;D) 9.16

Accepted Answer

The answer of If   is 7.15, then the...

Question 60

The magnitude of earthquake 1 is 5.9. If earthquake 2 is 13.49 times as intense as earth- quake 1, find the magnitude of earthquake 2. Round your answer to 1 decimal place.&#10;A) 7&#10;B) 4.8&#10;C) 19.4&#10;D) 8

Accepted Answer

The answer of The magnitude of earthquake 1 is 5.9....

Question 61

The size   , in thousands, of a certain animal population after   years is given in the table below. Use exponential regression to model the population size as a function of time.

Accepted Answer

The answer of The size   , in thousands,...

Question 62

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 63

Find an exponential model for the following data set.

Accepted Answer

The answer of Find an exponential model for the following...

Question 64

The size   , in thousands, of a certain animal population after   years is given in the table below. By what percentage can the population be expected to grow over any 6-year period?

Accepted Answer

The answer of The size   , in thousands,...

Deck 4: Exponential Functions