Question 1

What is comprised of the values of the dependent variable?
A)Range
B)Periodic function
C)Model
D)Domain

Accepted Answer

The range of a function consists of all the possible values of the dependent variable, based on the function's definition and the values of the independent variable(s).

Question 2

Solve the equation for the unknown quantity x.
-$2 ^ { 3 x } = 24$
A) $3.06$
B) $1.53$
C) $0.65$
D) $6.88$

Accepted Answer

To solve $2^{3x} = 24$, first express 24 as a power of 2. Since $24 = 2^3 \cdot 3$, and there's no simple way to express the entire right side as a power of 2, we approach it differently. We can rewrite the equation in logarithmic form or use the fact that $2^3 = 8$, but this doesn't directly help in simplifying $2^{3x} = 24$ in a straightforward manner. Instead, we recognize that the equation cannot be directly solved by expressing 24 as a power of 2. We use logarithms to solve for $x$:Given $2^{3x} = 24$, take the logarithm of both sides (using any base, but base 2 or 10 are most common). For simplicity and relevance to the choices given, we'll use the natural logarithm (ln), though the choice of logarithm base does not fundamentally change the approach:$$ln(2^{3x}) = ln(24)$$Using the logarithmic identity $ln(a^b) = b \cdot ln(a)$, we have:$$3x \cdot ln(2) = ln(24)$$Solving for $x$ gives:$$x = \frac{ln(24)}{3 \cdot ln(2)}$$Using a calculator to find the numerical value:$$x \approx 1.53$$Therefore, the correct answer is B, $1.53$.

Question 3

Exponential functions are useful for modeling enviromental issues.

Accepted Answer

Exponential functions can model a wide range of environmental issues, such as population growth, spread of diseases, and radioactive decay, due to their ability to represent rapid changes over time.

Question 4

Write an equation for the linear function and use it to answer the given question.
-In the town of Oak Forest, a 2% local sales tax and a 9% state sales tax are charged on all retail sales. Let p be the before-tax amount of a purchase in dollars. Let T be the after-tax amount of the
Purchase. Find a linear equation that describes how T varies with p. What is the total price of an
Item that costs $50 before taxes? 
A) $\mathrm { T } = 0.89 \mathrm { p } ; \$ 44.50$
B) $\mathrm { T } = 1.11 \mathrm { p } ; \$ 55.50$
C) $\mathrm { T } = 0.11 \mathrm { p } + 1 ; \$ 6.50$
D) $\mathrm { T } = 0.93 \mathrm { p } ; \$ 46.50$

Accepted Answer

The total sales tax is 2% + 9% = 11%, so the total cost after tax is 100% + 11% = 111% of the before-tax price, or 1.11p. For a $50 item, T = 1.11 * $50 = $55.50.

Question 5

There are currently 67 million cars in a certain country, decreasing by 0.5% annually. How many years will it take for this country to have 38 million cars? Round to the nearest year.
A)673 years
B)29 years
C)113 years
D)58 years

Accepted Answer

The answer of There are currently 67 million cars in...

Question 6

Provide an appropriate response.
-The population of a town with an initial population of 65,747 grows at a rate of 3.7% per year. Create a table showing the value of the quantity Q for the first 5 years or growth or decay. 
A)
$\begin{array}{cr}
\text { Year } & \text { Population } \
0 & 65,747 \
1 & 70,612 \
2 & 75,838 \
3 & 81,450 \
4 & 87,477 \
5 & 93,950
\end{array}$

B)
$\begin{array}{l}
\text { Year Population }\
\begin{array}{lr}
0 & 65,747 \
1 & 309,011 \
2 & 1,452,351 \
3 & 6,826,051 \
4 & 32,082,439 \
5 & 150,787,462
\end{array}
\end{array}$

C)
$\begin{array}{l}
\text { Year Population }\
\begin{array}{ll}
0 & 65,747 \
1 & 68,180 \
2 & 70,702 \
3 & 73,318 \
4 & 76,031 \
5 & 78,844
\end{array}
\end{array}$

D)
$\begin{array}{l}
\text { Year Population }\
\begin{array}{ll}
0 & 65,747 \
1 & 63,314 \
2 & 60,972 \
3 & 58,716 \
4 & 56,543 \
5 & 54,451
\end{array}
\end{array}$

Accepted Answer

The population grows at a rate of 3.7% per year, so each year's population is calculated by multiplying the previous year's population by 1.037. Option C correctly reflects this growth pattern.

Question 7

Solve the equation for the unknown quantity x.
-$3 ^ { x } = 27$
A) 3
B) $0.33$
C) $4.5$
D) 2

Accepted Answer

Since $27 = 3^3$, the equation $3^x = 27$ simplifies to $3^x = 3^3$, which means $x = 3$.

Question 8

What is comprised of the values of the independent variable?
A)Domain
B)Periodic function
C)Range
D)Model

Accepted Answer

The domain of a function consists of all the possible values of the independent variable for which the function is defined.

Question 9

Provide an appropriate response.
-What is the most compact mathematical representation of a function?
A)Graph
B)Data table
C)Equation
D)Domain

Accepted Answer

An equation provides the most compact and precise mathematical representation of a function, encapsulating its rule in a concise form.

Question 10

Find the slope of the graph and they y -intercept. Then sketch the graph for values of x between -10 and 10.
-$y = - 2 x + 1$
A) slope $= - \frac { 1 } { 2 } ; y$ intercept $= ( 0,1 )$

B) slope $= - 2 ; y$ intercept $= ( 0,1 )$

C) slope $= - 2 ; \mathrm { y }$ intercept $= ( 1,0 )$

D) slope $= - \frac { 1 } { 2 } ; y$ intercept $= ( 1,0 )$

Accepted Answer

The answer of Find the slope of the graph and...

Question 11

Create the required linear function an use it to answer the question.
-In 1995 the United States recovered 26% of its municipal solid wastes through recycling, up from 17% in 1990. Let P represent the percentage recycled and t the number of years since 1990. Based on
These two data points, create a linear equation for the function that describes how P varies as a Function of time. 
A) $P = 0.9 t - 17$
B) $P = 1.8 t + 17$
C) $P = - 1.8 t + 7$
D) $P = 1.8 t + 24$

Accepted Answer

The slope of the line is calculated as the change in percentage divided by the change in years: (26% - 17%) / (1995 - 1990) = 1.8. The equation of the line is P = 1.8t + 17, where 17 is the percentage in 1990.

Question 12

A certain drug is eliminated from the bloodstream exponentially with a half-life of 60 hours. Suppose that a patient receives an initial dose of 10 milligrams of the drug at midnight. Estimate
When the drug concentration will reach 20% of its initial level.
A)139 hours
B)199 hours
C)79 hours
D)23 hours

Accepted Answer

The drug's concentration decreases exponentially, following the formula $C = C_0 \times (1/2)^{t/T}$, where $C$ is the final concentration, $C_0$ is the initial concentration, $T$ is the half-life, and $t$ is the time in hours. To find when the concentration reaches 20% of its initial level, we set $C = 0.2C_0$ and solve for $t$, using $T = 60$ hours. This gives us $0.2 = (1/2)^{t/60}$. Solving for $t$, we find that $t \approx 139$ hours.

Question 13

After 60 years, a population growing at a rate of 4% per year will have grown by
twice as many people as a population growing at a rate of 2% per year.

Accepted Answer

The population growing at 4% per year will grow by 2^60 = 1,152,921,504,606,846,976 times its original size, while the population growing at 2% per year will grow by 2^60 = 1,152,921,504,606,846,976 times its original size.

Question 14

Provide an appropriate response.
-Cars sold  Crafty Bill's Cool Car Sales opened as a used car sales lot in 2001. The graph shows the number of cars sold as a function of time. What is the approximate number of cars sold in 2003?
A)550 cars
B)400 cars
C)500 cars
D)350 cars

Accepted Answer

The answer of Provide an appropriate response.
-Cars sold  Crafty...

Question 15

The population of a town with an initial population of 60,196 is decreasing at a rate of 4.2% per year. Create a table showing the value of the quantity Q for the first 5 years or growth or decay. 
A)
$\begin{array}{cr}
\text { Year } & \text { Population } \
0 & 60,196 \
1 & 55,140 \
2 & 50,508 \
3 & 46,265 \
4 & 42,379 \
5 & 38,819
\end{array}$

B)
$\begin{array}{cr}
\text { Year } & \text { Population } \
0 & 60,196 \
1 & 58,932 \
2 & 57,694 \
3 & 56,483 \
4 & 55,297 \
5 & 54,135
\end{array}$

C)
$\begin{array}{cr}
\text { Year } & \text { Population } \
0 & 60,196 \
1 & 57,668 \
2 & 55,246 \
3 & 52,925 \
4 & 50,703 \
5 & 48,573
\end{array}$

D)
$\begin{array}{l}
\text { Year Population }\
\begin{array}{ll}
0 & 60,196 \
1 & 62,724 \
2 & 65,359 \
3 & 68,104 \
4 & 70,964 \
5 & 73,945
\end{array}
\end{array}$

Accepted Answer

The population decreases by 4.2% each year, so the population for each subsequent year is calculated by multiplying the previous year's population by 0.958 (100% - 4.2% = 95.8% = 0.958). Option C correctly applies this calculation.

Question 16

Write an equation for the linear function and use it to answer the given question.
-You can purchase a motorcycle for $7101 or lease it for a down payment of $783 and $243 per month. Find an equation that describes how the cost of the lease depends on time. How long can
You lease the motorcycle before you've paid more than its purchase price. 
A) $\mathrm { L } = 783 + 243 \mathrm {~m} ; 26$ months
B) $L = 783 + 243 \mathrm {~m} ; 32$ months
C) $L = 783 - 243 m ; 32$ months
D) $\mathrm { L } = 783 - 243 \mathrm {~m} ; 26$ months

Accepted Answer

The cost of leasing the motorcycle is represented by the equation $L = 783 + 243m$, where $783$ is the down payment, $243$ is the monthly payment, and $m$ is the number of months. To find when the lease cost exceeds the purchase price, set $7101 = 783 + 243m$ and solve for $m$, which gives $m = 26$ months.

Question 17

Provide an appropriate response.
-The population of a town with an initial population of 64,546 is decreasing at a rate of 3.7% per year. Create an exponential function of the form Q
 $\mathrm { Q } = \mathrm { Q } _ { 0 } \times ( 1 + \mathrm { r } ) ^ { \mathrm { t } }$ 
A) $\mathrm { Q } = 64,546 \times ( 1.037 ) ^ { \mathrm { t } }$
B) $Q = 64,546 \times ( 0.963 ) ^ { t }$
C) $Q = 64,546 \times ( - 2.7 ) ^ { t }$
D) $\mathrm { Q } = 64,546 \times ( 1.074 ) ^ { \mathrm { t } }$

Accepted Answer

The correct formula for a decreasing population is $Q = Q_0 \times (1 - r)^t$, where $r$ is the rate of decrease. Since the population is decreasing by 3.7%, $r = 0.037$, and thus $1 - r = 0.963$. Therefore, the correct formula is $Q = 64,546 \times (0.963)^t$.

Question 18

Provide an appropriate response.
-The following graph represents a function. Describe the function in words. 
A)The function shows a steadily increasing average age of death between 1980 and 2004.
B)The function shows a steadily decreasing average age of death between 1980 and 2004.
C)The function shows that the average age of death increases between 1980 and 1992 and does not change between 1992 and 2004.
D)The function shows that the average age of death per year does not change between 1980 and 2004.

Accepted Answer

The answer of Provide an appropriate response.
-The following graph represents...

Question 19

Write an equation for the linear function and use it to answer the given question.
-Normaltown High School's pool record for the 100-yard freestyle was 47.8 in 1990. Assume that the record falls at a constant rate of 0.03 second per year. What does the model predict for the
Record in 2010? 
A) $R = 47.8 t + 0.03 ; 956.03$ seconds
B) $\mathrm { R } = 47.8 \mathrm { t } - 0.03 ; 955.97$ seconds
C) $\mathrm { R } = 47.8 + 0.03 \mathrm { t } ; 48.40$ seconds
D) $R = 47.8 - 0.03 \mathrm { t } ; 47.20$ seconds

Accepted Answer

The correct equation to model the situation is $R = 47.8 - 0.03t$, where $t$ is the number of years since 1990. For 2010, $t = 20$, so the record would be $47.8 - 0.03(20) = 47.20$ seconds.

Question 20

Create the required linear function an use it to answer the question.
-Suppose that you were 22 inches long at birth and 3.5 feet tall on your tenth birthday. Based on these two data points, create a linear equation for the function that describes how height varies
With age. Use the equation to predict the height at age 2 and 41. 
A) $h = 22 + 6.4 a$; 26 inches; $284.4$ inches
B) $h = 22 + 2 a$; 26 inches; 104 inches
C) $h = 22 + 10.6 a ; 34.4$ inches; $456.6$ inches
D) $h = 22 - 2 a$; 18 inches; $- 60$ inches

Accepted Answer

First, convert the height at age 10 from feet to inches: 3.5 feet = 42 inches. The change in height from birth to age 10 (22 inches to 42 inches) is 20 inches over 10 years, so the rate of change is 20 inches / 10 years = 2 inches per year. The linear equation is $h = 22 + 2a$, where $h$ is the height in inches and $a$ is the age in years. At age 2, $h = 22 + 2(2) = 26$ inches. At age 41, $h = 22 + 2(41) = 104$ inches.

Quiz 9: Modeling Our World

What is comprised of the values of the dependent variable?

Solve the equation for the unknown quantity x. - $2 ^ { 3 x } = 24$

Exponential functions are useful for modeling enviromental issues.

There are currently 67 million cars in a certain country, decreasing by 0.5% annually. How many years will it take for this country to have 38 million cars? Round to the nearest year.

Provide an appropriate response. -The population of a town with an initial population of 65,747 grows at a rate of 3.7% per year. Create a table showing the value of the quantity Q for the first 5 years or growth or decay.

Solve the equation for the unknown quantity x. - $3 ^ { x } = 27$

What is comprised of the values of the independent variable?

Provide an appropriate response. -What is the most compact mathematical representation of a function?

Find the slope of the graph and they y -intercept. Then sketch the graph for values of x between -10 and 10. - $y = - 2 x + 1$

A certain drug is eliminated from the bloodstream exponentially with a half-life of 60 hours. Suppose that a patient receives an initial dose of 10 milligrams of the drug at midnight. Estimate When the drug concentration will reach 20% of its initial level.

After 60 years, a population growing at a rate of 4% per year will have grown by twice as many people as a population growing at a rate of 2% per year.

Provide an appropriate response. -Cars sold Crafty Bill's Cool Car Sales opened as a used car sales lot in 2001. The graph shows the number of cars sold as a function of time. What is the approximate number of cars sold in 2003?

The population of a town with an initial population of 60,196 is decreasing at a rate of 4.2% per year. Create a table showing the value of the quantity Q for the first 5 years or growth or decay.

Provide an appropriate response. -The population of a town with an initial population of 64,546 is decreasing at a rate of 3.7% per year. Create an exponential function of the form Q $\mathrm { Q } = \mathrm { Q } _ { 0 } \times ( 1 + \mathrm { r } ) ^ { \mathrm { t } }$

Provide an appropriate response. -The following graph represents a function. Describe the function in words.

Write an equation for the linear function and use it to answer the given question. -Normaltown High School's pool record for the 100-yard freestyle was 47.8 in 1990. Assume that the record falls at a constant rate of 0.03 second per year. What does the model predict for the Record in 2010?

Quiz 9: Modeling Our World

What is comprised of the values of the dependent variable?

Solve the equation for the unknown quantity x. - 23x=242 ^ { 3 x } = 2423x=24

Exponential functions are useful for modeling enviromental issues.

There are currently 67 million cars in a certain country, decreasing by 0.5% annually. How many years will it take for this country to have 38 million cars? Round to the nearest year.

Provide an appropriate response. -The population of a town with an initial population of 65,747 grows at a rate of 3.7% per year. Create a table showing the value of the quantity Q for the first 5 years or growth or decay.

Solve the equation for the unknown quantity x. - 3x=273 ^ { x } = 273x=27

What is comprised of the values of the independent variable?

Provide an appropriate response. -What is the most compact mathematical representation of a function?

Find the slope of the graph and they y -intercept. Then sketch the graph for values of x between -10 and 10. - y=−2x+1y = - 2 x + 1y=−2x+1

A certain drug is eliminated from the bloodstream exponentially with a half-life of 60 hours. Suppose that a patient receives an initial dose of 10 milligrams of the drug at midnight. Estimate When the drug concentration will reach 20% of its initial level.

After 60 years, a population growing at a rate of 4% per year will have grown by twice as many people as a population growing at a rate of 2% per year.

Provide an appropriate response. -Cars sold Crafty Bill's Cool Car Sales opened as a used car sales lot in 2001. The graph shows the number of cars sold as a function of time. What is the approximate number of cars sold in 2003?

The population of a town with an initial population of 60,196 is decreasing at a rate of 4.2% per year. Create a table showing the value of the quantity Q for the first 5 years or growth or decay.

Provide an appropriate response. -The population of a town with an initial population of 64,546 is decreasing at a rate of 3.7% per year. Create an exponential function of the form Q Q=Q0×(1+r) t\mathrm { Q } = \mathrm { Q } _ { 0 } \times ( 1 + \mathrm { r } ) ^ { \mathrm { t } }Q=Q0​×(1+r) t

Provide an appropriate response. -The following graph represents a function. Describe the function in words.

Write an equation for the linear function and use it to answer the given question. -Normaltown High School's pool record for the 100-yard freestyle was 47.8 in 1990. Assume that the record falls at a constant rate of 0.03 second per year. What does the model predict for the Record in 2010?

Solve the equation for the unknown quantity x. - $2 ^ { 3 x } = 24$

Solve the equation for the unknown quantity x. - $3 ^ { x } = 27$

Find the slope of the graph and they y -intercept. Then sketch the graph for values of x between -10 and 10. - $y = - 2 x + 1$

Provide an appropriate response. -The population of a town with an initial population of 64,546 is decreasing at a rate of 3.7% per year. Create an exponential function of the form Q $\mathrm { Q } = \mathrm { Q } _ { 0 } \times ( 1 + \mathrm { r } ) ^ { \mathrm { t } }$