Question 1

The table below is filled correctly. $$\begin{array} { | l | l | } 
\hline \text { Exponential form } & \text { Logarithmic form } \
\hline 5 ^ { 4 } = 625 & \log _ { 5 } 625 = 4 \
\hline 0.6 ^ { 2 } = 0.36 & \log _ { 0.6 } 0.36 = 2 \
\hline 7 ^ { 0 } = 1 & \log _ { 7 } 1 = 0 \
\hline 8 ^ { - 1 } = 0.125 & \log _ { 8 } 0.125 = - 1 \
\hline
\end{array}$$

Accepted Answer

Each logarithmic form correctly represents the exponential form given, adhering to the definition of logarithms.

Question 2

Model the data using an exponential function $$f ( x ) = A b ^ { x }$$ . $$\begin{array} { | l | l | l | l | } 
\hline \boldsymbol { x } & 0 & 1 & 2 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 350 & 175 & 87.5 \
\hline
\end{array}$$   Select the correct answer.

Accepted Answer

The exponential function that models the data is $f(x) = 350(0.5)^x$. This is because when $x=0$, $f(x)=350$, and for each increase in $x$ by 1, $f(x)$ is halved, which matches the behavior of multiplying by $0.5$.

Question 3

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 40 grams of plutonium-239 to decay to 3 grams 
 
Round your answer to the nearest hundreds.
 
__________ years

Accepted Answer

The answer of Plutonium-239 is used as a fuel for...

Question 4

Model the data using an exponential function $$f ( x ) = A b ^ { x }$$ . $$\begin{array} { | l | l | l | } 
\hline \boldsymbol { x } & 1 & 2 \
\hline f ( x ) & 13 & 16.9 \
\hline
\end{array}$$   Select the correct answer.

Accepted Answer

To find the correct model, we use the given data points (1, 13) and (2, 16.9) in the exponential function $f(x) = Ab^x$. For $x = 1$, $f(1) = 13 = Ab^1 = Ab$. For $x = 2$, $f(2) = 16.9 = Ab^2$.Dividing the second equation by the first gives $\frac{16.9}{13} = b$, which simplifies to $b = 1.3$.Substituting $b = 1.3$ into $f(1) = 13 = A(1.3)^1$ gives $A = 10$.Therefore, the correct model is $f(x) = 10(1.3)^x$.

Question 5

Find the associated exponential decay model.
​ $Q = 6,000$ when $t = 0$ ; Half-life = 5

Accepted Answer

The answer of Find the associated exponential decay model.
​ \(Q...

Question 6

How long, to the nearest year, will it take an investment in Canada to double its value if the interest is compounded every six months  Please round the answer to the nearest year. $$\begin{array} { | l | l | l | l | l | l | } 
\hline \text { Country } & \text { U.S. } & \text { Japan } & \text { Canada } & \text { Korea } & \text { Australia } \
\hline \text { Yield } & 5.3 \% & 1.5 \% & 5.2 \% & 5.4 \% & 6.0 \% \
\hline
\end{array}$$  
t = __________ year(s)

Accepted Answer

The answer of How long, to the nearest year, will...

Question 7

Rock Solid Bank & Trust is offering a CD that pays 5% compounded continuously. How much interest would a $1,000 deposit earn over 12 years  (Round your answer to the nearest dollar.)  
Select the correct answer.

Accepted Answer

The formula to find the amount of interest earned over time with continuous compounding is A=Pe^(rt), where P is the principal, r is the interest rate, t is the time in years, and e is the constant 2.71828. Plugging in the given values, we get A=1000e^(0.05*12). Solving for A, we get approximately $1834.67. Subtracting the principal, we get the interest earned: $1834.67 - $1000 = $834.67, which rounds to $822 to the nearest dollar. Therefore, the correct answer is B) $822.

Question 8

Which of the following five functions will be largest for large values of x   
Select the correct answer.

Accepted Answer

For large values of $x$, exponential functions (like $8^x$) grow faster than polynomial functions, regardless of the polynomial's degree or coefficients.

Question 9

Given the graph of the functions $f _ { 1 } ( x ) = 2.5 ^ { x }$ and $f _ { 2 } ( x ) = 2.8 ^ { x }$ . Identify which graph corresponds to $f _ { 2 } ( x ) = 2.8 ^ { x }$ .     
Select the correct answer.

Accepted Answer

The answer of Given the graph of the functions \(f...

Question 10

The U.S. population was 170 million in 1950 and 240 million in 1990. Assuming exponential population growth, what will the population be in the year 2020  Round your answer to the nearest million.  
Select the correct answer.

Accepted Answer

The exponential growth formula is $P = P_0e^{rt}$, where $P_0$ is the initial population, $r$ is the rate of growth, and $t$ is the time in years. Given $P_0 = 170$ million in 1950 and $P = 240$ million in 1990, we can solve for $r$ since $t = 40$ years. Then, using $r$ and $P_0 = 240$ million, we predict the population for 2020, which is 30 years from 1990. The calculation yields approximately 311 million.

Question 11

The table lists interest rates on long-term investments (based on 10-year government bonds) in several countries in 2004-2005. Assuming that you invest $12,000 in Japan, how long (to the nearest year) must you wait before your investment is worth $18,000 if the interest is compounded annually  $$\begin{array} { | l | l | l | l | } 
\hline \text { Country } & \text { U.S. } & \text { Japan } & \text { Canada } \
\hline \text { Yield } & 5.3 \% & 1.5 \% & 5.2 \% \
\hline
\end{array}$$  
__________ year(s)

Accepted Answer

The answer of The table lists interest rates on long-term...

Question 12

Soon after taking an aspirin, a patient has absorbed 310 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially with half being removed every 2 hours, find the amount of aspirin in the bloodstream after 9 hours. ​
Select the correct answer.
​

Accepted Answer

The amount of aspirin in the bloodstream after a certain number of hours can be calculated using the formula for exponential decay: A = P(1/2)^(t/h), where A is the amount after time t, P is the initial amount, and h is the half-life. Here, P = 310 mg, t = 9 hours, and h = 2 hours. Plugging these values into the formula gives A = 310(1/2)^(9/2) ≈ 13.7 mg.

Question 13

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.
​ $f ( t ) = 23.7 ( 0.996 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$ ​ $Q _ { 0 } =\_\_\_\_$ ​ $k =\_\_\_\_$

Accepted Answer

The answer of Convert the exponential function to the form...

Question 14

Graph the function.   $$f ( x ) = 4 \left( 2 ^ { - x } \right)$$ 
Select the correct answer.

Accepted Answer

The answer of Graph the function.   \[f (...

Question 15

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $$\begin{array} { | l | l | l | l | } 
\hline \boldsymbol { x } & 0 & 1 & 2 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 40 & 80 & 160 \
\hline
\end{array}$$  Select the correct answer.

Accepted Answer

The exponential function that models the data is $f(x) = 40(2)^x$. This is because when $x=0$, $f(x)=40$, which matches the initial value $A=40$, and the rate of increase is by a factor of 2 for each increment in $x$, indicating $b=2$.

Question 16

How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 15% per year 
 
Round the answer to the nearest year.
 
__________ years

Accepted Answer

The answer of How long, to the nearest year, will...

Question 17

The table below is filled correctly. $$\begin{array} { | l | l | } 
\hline \text { Exponential form } & \text { Logarithmic form } \
\hline 5 ^ { 1 } = 5 & \log _ { 5 } 5 = 1 \
\hline 7 ^ { 4 } = 2,401 & \log _ { 7 } 2,401 = 4 \
\hline 9 ^ { - 1 } = \frac { 1 } { 9 } & \log _ { 9 } \frac { 1 } { 9 } = - 1 \
\hline 10 ^ { 3 } = 1,000 & \log _ { 10 } 1,000 = 3 \
\hline
\end{array}$$

Accepted Answer

The logarithmic form correctly represents the exponential form for each given pair, following the definition of logarithms.

Question 18

Find an equation for an exponential function that passes through the pair of points $( 4,3 )$ and $( 7,1 )$ .   $y = A b ^ { x } ( b > 0 )$

Accepted Answer

By substituting the points into the exponential function and solving the system of equations, the values of $A$ and $b$ are found to be approximately 12.9802 and 0.6934, respectively.

Question 19

Given the graph of the functions $f _ { 1 } ( x ) = 2.3 ^ { - x }$ and $f _ { 2 } ( x ) = e ^ { - x }$ . Determine the color of the graph that corresponds to $f _ { 1 } ( x )$ .     
Select the correct answer.

Accepted Answer

The answer of Given the graph of the functions \(f...

Question 20

The table is filled correctly. $$\begin{array} { | l | l | } 
\hline \text { Exponential form } & \text { Logarithmic form } \
\hline 8 ^ { - 3 } = 0.001953 & \log _ { 8 } 0.001953 = 3 \
\hline 2 ^ { 0 } = 1 & \log _ { 2 } 1 = 0 \
\hline 0.4 ^ { 4 } = 0.0256 & \log _ { 0.4 } 0.0256 = 4 \
\hline 0.8 ^ { - 3 } = 1.953125 & \log _ { 0.8 } 1.953125 = 3 \
\hline
\end{array}$$

Accepted Answer

The logarithmic form corresponding to $8^{-3} = 0.001953$ should be $\log_{8} 0.001953 = -3$, not $3$.

Quiz 2: Nonlinear Functions and Models

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 350 & 175 & 87.5 \\\hline\end{array}$ Select the correct answer.

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 40 grams of plutonium-239 to decay to 3 grams Round your answer to the nearest hundreds. __________ years

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | } \hline \boldsymbol { x } & 1 & 2 \\\hline f ( x ) & 13 & 16.9 \\\hline\end{array}$ Select the correct answer.

Find the associated exponential decay model. $Q = 6,000$ when $t = 0$ ; Half-life = 5

Rock Solid Bank & Trust is offering a CD that pays 5% compounded continuously. How much interest would a $1,000 deposit earn over 12 years (Round your answer to the nearest dollar.) Select the correct answer.

Which of the following five functions will be largest for large values of x Select the correct answer.

The U.S. population was 170 million in 1950 and 240 million in 1990. Assuming exponential population growth, what will the population be in the year 2020 Round your answer to the nearest million. Select the correct answer.

Soon after taking an aspirin, a patient has absorbed 310 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially with half being removed every 2 hours, find the amount of aspirin in the bloodstream after 9 hours. Select the correct answer.

Convert the exponential function to the form indicated. Round all coefficients to four significant digits. $f ( t ) = 23.7 ( 0.996 ) ^ { t }$ ; $f ( t ) = Q _ { 0 } e ^ { - k ^ { \prime } t }$ $Q _ { 0 } =\_\_\_\_$ $k =\_\_\_\_$

Graph the function. $f ( x ) = 4 \left( 2 ^ { - x } \right)$ Select the correct answer.

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 40 & 80 & 160 \\\hline\end{array}$ Select the correct answer.

How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 15% per year Round the answer to the nearest year. __________ years

Find an equation for an exponential function that passes through the pair of points $( 4,3 )$ and $( 7,1 )$ . $y = A b ^ { x } ( b > 0 )$

Quiz 2: Nonlinear Functions and Models

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 40 grams of plutonium-239 to decay to 3 grams Round your answer to the nearest hundreds. __________ years

Model the data using an exponential function f(x) =Abxf ( x ) = A b ^ { x }f(x) =Abx . x12f(x) 1316.9\begin{array} { | l | l | l | } \hline \boldsymbol { x } & 1 & 2 \\\hline f ( x ) & 13 & 16.9 \\\hline\end{array}xf(x) ​113​216.9​​ Select the correct answer.

Find the associated exponential decay model. ​ Q=6,000Q = 6,000Q=6,000 when t=0t = 0t=0 ; Half-life = 5

Rock Solid Bank & Trust is offering a CD that pays 5% compounded continuously. How much interest would a $1,000 deposit earn over 12 years (Round your answer to the nearest dollar.) Select the correct answer.

Which of the following five functions will be largest for large values of x Select the correct answer.

Given the graph of the functions f1(x) =2.5xf _ { 1 } ( x ) = 2.5 ^ { x }f1​(x) =2.5x and f2(x) =2.8xf _ { 2 } ( x ) = 2.8 ^ { x }f2​(x) =2.8x . Identify which graph corresponds to f2(x) =2.8xf _ { 2 } ( x ) = 2.8 ^ { x }f2​(x) =2.8x . Select the correct answer.

The U.S. population was 170 million in 1950 and 240 million in 1990. Assuming exponential population growth, what will the population be in the year 2020 Round your answer to the nearest million. Select the correct answer.

Soon after taking an aspirin, a patient has absorbed 310 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially with half being removed every 2 hours, find the amount of aspirin in the bloodstream after 9 hours. ​ Select the correct answer. ​

Graph the function. f(x) =4(2−x) f ( x ) = 4 \left( 2 ^ { - x } \right) f(x) =4(2−x) Select the correct answer.