Question
A fuel is purified by passing it through a clay pipe. Each foot of the clay pipe removes a fixed percentage of impurities in the fuel. Let f(x) represent the amount of pollutants, in tons, left in the fuel after it had passed through x feet of the clay pipe:
f(x) = 10(0.8)x
What does the number 0.8 represent?
Every foot of the pipe keeps 80% of the pollutant.
Every foot of the pipe removes 80% of the pollutant.
The original amount of pollutant present in the fuel was 80 tons.
The original amount of pollutant present in the fuel was 20 tons.
Question 2(Multiple Choice Worth 1 points)
(03.05 MC)
Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $8,539.95 $10,751.61
2000 $13,047.20 $13,722.08
2005 $17,554.45 $17,577.05
2006 $18,455.90 $18,455.90
2007 $19,357.35 $19,378.70
2010 $22,061.70 $22,433.27
f(x) is exponential; an exponential function increases more slowly than a linear function.
f(x) is exponential; f(x) increased more overall than g(x).
g(x) is exponential; an exponential function increases faster than a linear function.
g(x) is exponential; g(x) has a higher starting value and higher ending value.
Question 3(Multiple Choice Worth 1 points)
(03.05 MC)
Graph f(x) = 46(0.75)x. What is the constant percent rate of change of f(x) with respect to x? Does the graph represent growth or decay?
75% growth
75% decay
25% growth
25% decay
Question 4(Multiple Choice Worth 1 points)
(03.03 LC)
Given the exponential function f(x) = 16(0.75)x, classify the function as exponential growth or decay and determine the percent rate of growth or decay.
Exponential growth, 75% increase
Exponential decay, 75% decrease
Exponential growth, 25% increase
Exponential decay, 25% decrease
Question 5(Multiple Choice Worth 1 points)
(03.03 MC)
The number of bacteria in a sample increased by a factor of 2 every minute from 9:05 a.m. to 9:55 a.m. The function below shows the number of bacteria in the sample f(x) after x minutes from 9:05 a.m:
f(x) = 10(2)x
Which of the following is a reasonable domain for the function?
0 ≤ x ≤ 10
5 ≤ x ≤ 55
0 ≤ x ≤ 50
All positive integers
Question 6(Multiple Choice Worth 1 points)
(03.05 MC)
Albert invested money into the stock market, and the table represents his earnings. What type of function could be used to model his bank account as a function of time? Justify your answer.
Week Balance ($)
1 428
2 620
3 812
4 1,004
This is an exponential function because there is a common difference in the balance between the weeks.
This is an exponential function because there is a common ratio in the balance between the weeks.
This is a linear function because there is a common difference in the balance between the weeks.
This is a linear function because there is a common ratio in the balance between the weeks.
Question 7(Multiple Choice Worth 1 points)
(03.02 MC)
Initially, there were only 86 weeds in the garden. The weeds grew at a rate of 8% each week. The following function represents the weekly weed growth: f(x) = 86(1.08)x. Rewrite the function to show how quickly the weeds grow each day.
f(x) = 86(1.08)7x; grows approximately at a rate of 5.6% daily
f(x) = 86(1.087)x; grows approximately at a rate of 0.56% daily
f(x) = 86(1.01)x; grows approximately at a rate of 0.1% daily
f(x) = 86(1.01)7x; grows approximately at a rate of 1% daily
Question 8(Multiple Choice Worth 1 points)
(03.01 LC)
Simplify the square root of 5 times the cube root of 5.
five to the five sixths power
five to the one sixth power
five to the two thirds power
five to the seven sixths power
Question 9(Multiple Choice Worth 1 points)
(03.04 MC)
Does the function y equals 14 times the quantity 1 over 2 end quantity to the x power represent exponential growth or decay? What is the equation of the asymptote?
Decay; y = 0
Growth; y = 0
Decay; x = −3
Growth; x = −3
Question 10(Multiple Choice Worth 1 points)
(03.02 MC)
The price of bronze has increased by 10% per year from 2000. In the year 2000, Harry bought a bronze medal for $120. Which of the following equations can be used to represent the price of the medal x years after 2000?
y = 120(0.9)x
y = 10(0.9)x
y = 120(1.1)x
y = 10(1.1)x
Question 11(Multiple Choice Worth 1 points)
(03.04 MC)
The population f(x), in millions, of State A of a country after x years is represented by the function shown below:
f(x) = (1.08)x
The graph shows the population g(x), in millions, of State B of the country after x years:
graph of exponential function g of x that curves up from left to right and goes through points 0 comma 2 and 9 comma 4
Which conclusion is correct about the populations of State A and State B?
The original population of State B was half the original population of State A.
The original population of State A was half of the original population of State B.
The original population of State B was one-fourth the original population of State A.
The original population of State A was one-fourth the original population of State B.
Question 12(Multiple Choice Worth 1 points)
(03.04 LC)
Graph the function f(x) = 18(0.8)x.
an exponential graph decreasing from the left and crossing the y axis at 0 comma 21
an exponential graph decreasing from the left and crossing the y axis at 0 comma 5
an exponential graph decreasing from the left and crossing the y axis at 0 comma 18
an exponential graph decreasing from the left and crossing the y axis at 0 comma 8
Question 13(Multiple Choice Worth 1 points)
(03.03 MC)
Use the graph representing bacteria decay to estimate the domain of the function and solve for the average rate of change across the domain.
An exponential function titled Bacteria Decay with x axis labeled Time, in Minutes, and y axis labeled Amount of Bacteria, in Thousands, decreasing to the right with a y intercept of 0 comma 60 and an x intercept of 18 comma 0.
0 ≤ x ≤ 18, −3.33
0 ≤ x ≤ 18, −0.3
0 ≤ y ≤ 60, −3.33
0 ≤ y ≤ 60, −0.3
Question 14(Multiple Choice Worth 1 points)
(03.02 LC)
The function f(x) = 3,267(1 + 0.02)x represents the amount of money in a savings account where x represents time in years. What does the 3,267 represent?
The amount of money in the savings account after one year
The amount of money added to the savings account each year
The initial amount of money placed in the savings account
The growth rate
Question 15(Multiple Choice Worth 1 points)
(03.01 MC)
An expression is shown below:
the square root of 18 plus the square root of 2
Which statement is true about the expression?
It is rational and equal to 4 times the square root of 2.
It is irrational and equal to 4 times the square root of 2.
It is rational and equal to 3 times the square root of 2.
It is irrational and equal to 3 times the square root of 2.
f(x) = 10(0.8)x
What does the number 0.8 represent?
Every foot of the pipe keeps 80% of the pollutant.
Every foot of the pipe removes 80% of the pollutant.
The original amount of pollutant present in the fuel was 80 tons.
The original amount of pollutant present in the fuel was 20 tons.
Question 2(Multiple Choice Worth 1 points)
(03.05 MC)
Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $8,539.95 $10,751.61
2000 $13,047.20 $13,722.08
2005 $17,554.45 $17,577.05
2006 $18,455.90 $18,455.90
2007 $19,357.35 $19,378.70
2010 $22,061.70 $22,433.27
f(x) is exponential; an exponential function increases more slowly than a linear function.
f(x) is exponential; f(x) increased more overall than g(x).
g(x) is exponential; an exponential function increases faster than a linear function.
g(x) is exponential; g(x) has a higher starting value and higher ending value.
Question 3(Multiple Choice Worth 1 points)
(03.05 MC)
Graph f(x) = 46(0.75)x. What is the constant percent rate of change of f(x) with respect to x? Does the graph represent growth or decay?
75% growth
75% decay
25% growth
25% decay
Question 4(Multiple Choice Worth 1 points)
(03.03 LC)
Given the exponential function f(x) = 16(0.75)x, classify the function as exponential growth or decay and determine the percent rate of growth or decay.
Exponential growth, 75% increase
Exponential decay, 75% decrease
Exponential growth, 25% increase
Exponential decay, 25% decrease
Question 5(Multiple Choice Worth 1 points)
(03.03 MC)
The number of bacteria in a sample increased by a factor of 2 every minute from 9:05 a.m. to 9:55 a.m. The function below shows the number of bacteria in the sample f(x) after x minutes from 9:05 a.m:
f(x) = 10(2)x
Which of the following is a reasonable domain for the function?
0 ≤ x ≤ 10
5 ≤ x ≤ 55
0 ≤ x ≤ 50
All positive integers
Question 6(Multiple Choice Worth 1 points)
(03.05 MC)
Albert invested money into the stock market, and the table represents his earnings. What type of function could be used to model his bank account as a function of time? Justify your answer.
Week Balance ($)
1 428
2 620
3 812
4 1,004
This is an exponential function because there is a common difference in the balance between the weeks.
This is an exponential function because there is a common ratio in the balance between the weeks.
This is a linear function because there is a common difference in the balance between the weeks.
This is a linear function because there is a common ratio in the balance between the weeks.
Question 7(Multiple Choice Worth 1 points)
(03.02 MC)
Initially, there were only 86 weeds in the garden. The weeds grew at a rate of 8% each week. The following function represents the weekly weed growth: f(x) = 86(1.08)x. Rewrite the function to show how quickly the weeds grow each day.
f(x) = 86(1.08)7x; grows approximately at a rate of 5.6% daily
f(x) = 86(1.087)x; grows approximately at a rate of 0.56% daily
f(x) = 86(1.01)x; grows approximately at a rate of 0.1% daily
f(x) = 86(1.01)7x; grows approximately at a rate of 1% daily
Question 8(Multiple Choice Worth 1 points)
(03.01 LC)
Simplify the square root of 5 times the cube root of 5.
five to the five sixths power
five to the one sixth power
five to the two thirds power
five to the seven sixths power
Question 9(Multiple Choice Worth 1 points)
(03.04 MC)
Does the function y equals 14 times the quantity 1 over 2 end quantity to the x power represent exponential growth or decay? What is the equation of the asymptote?
Decay; y = 0
Growth; y = 0
Decay; x = −3
Growth; x = −3
Question 10(Multiple Choice Worth 1 points)
(03.02 MC)
The price of bronze has increased by 10% per year from 2000. In the year 2000, Harry bought a bronze medal for $120. Which of the following equations can be used to represent the price of the medal x years after 2000?
y = 120(0.9)x
y = 10(0.9)x
y = 120(1.1)x
y = 10(1.1)x
Question 11(Multiple Choice Worth 1 points)
(03.04 MC)
The population f(x), in millions, of State A of a country after x years is represented by the function shown below:
f(x) = (1.08)x
The graph shows the population g(x), in millions, of State B of the country after x years:
graph of exponential function g of x that curves up from left to right and goes through points 0 comma 2 and 9 comma 4
Which conclusion is correct about the populations of State A and State B?
The original population of State B was half the original population of State A.
The original population of State A was half of the original population of State B.
The original population of State B was one-fourth the original population of State A.
The original population of State A was one-fourth the original population of State B.
Question 12(Multiple Choice Worth 1 points)
(03.04 LC)
Graph the function f(x) = 18(0.8)x.
an exponential graph decreasing from the left and crossing the y axis at 0 comma 21
an exponential graph decreasing from the left and crossing the y axis at 0 comma 5
an exponential graph decreasing from the left and crossing the y axis at 0 comma 18
an exponential graph decreasing from the left and crossing the y axis at 0 comma 8
Question 13(Multiple Choice Worth 1 points)
(03.03 MC)
Use the graph representing bacteria decay to estimate the domain of the function and solve for the average rate of change across the domain.
An exponential function titled Bacteria Decay with x axis labeled Time, in Minutes, and y axis labeled Amount of Bacteria, in Thousands, decreasing to the right with a y intercept of 0 comma 60 and an x intercept of 18 comma 0.
0 ≤ x ≤ 18, −3.33
0 ≤ x ≤ 18, −0.3
0 ≤ y ≤ 60, −3.33
0 ≤ y ≤ 60, −0.3
Question 14(Multiple Choice Worth 1 points)
(03.02 LC)
The function f(x) = 3,267(1 + 0.02)x represents the amount of money in a savings account where x represents time in years. What does the 3,267 represent?
The amount of money in the savings account after one year
The amount of money added to the savings account each year
The initial amount of money placed in the savings account
The growth rate
Question 15(Multiple Choice Worth 1 points)
(03.01 MC)
An expression is shown below:
the square root of 18 plus the square root of 2
Which statement is true about the expression?
It is rational and equal to 4 times the square root of 2.
It is irrational and equal to 4 times the square root of 2.
It is rational and equal to 3 times the square root of 2.
It is irrational and equal to 3 times the square root of 2.
Answers
GPT-4o mini
Let's go through each question one by one and provide the correct answers:
**Question 1:**
The function \( f(x) = 10(0.8)^x \) represents the amount of pollution left after passing through \( x \) feet of the clay pipe. The number 0.8 represents:
- **Correct Answer**: Every foot of the pipe keeps 80% of the pollutant.
**Question 2:**
Given the profits of the two businesses, we need to determine which function is exponential by observing the profit values:
- **Correct Answer**: f(x) is exponential; f(x) increased more overall than g(x).
**Question 3:**
The function \( f(x) = 46(0.75)^x \) indicates decay because 0.75 signifies a reduction per unit increase in \( x \):
- **Correct Answer**: 75% decay.
**Question 4:**
The function \( f(x) = 16(0.75)^x \) indicates decay since the base is less than 1:
- **Correct Answer**: Exponential decay, 25% decrease. (The decay per time period is 1 - 0.75 = 0.25, or 25%.)
**Question 5:**
The bacteria doubling every minute is expressed as:
- **Correct Answer**: 0 ≤ x ≤ 50 (from 9:05 a.m. to 9:55 a.m., it totals 50 minutes).
**Question 6:**
The balance increase from week to week reflects a non-linear relationship:
- **Correct Answer**: This is a linear function because there is a common difference in the balance between the weeks.
**Question 7:**
To convert weekly growth to daily growth, we need to adjust the base as follows:
- **Correct Answer**: \( f(x) = 86(1.01)^{7x} \); grows approximately at a rate of 1% daily. (Daily rate can be estimated from \( (1.08)^{\frac{1}{7}} - 1 \approx 0.0101 \)).
**Question 8:**
Simplifying the expression involves use of fractional exponents:
- **Correct Answer**: five to the seven sixths power.
**Question 9:**
The function \( y = 14 \cdot (1/2)^x \) indicates decay and the equation of the asymptote:
- **Correct Answer**: Decay; y = 0.
**Question 10:**
For the price increase of 10% per year:
- **Correct Answer**: \( y = 120(1.1)^x \).
**Question 11:**
The initial population comparison:
- **Correct Answer**: The original population of State B was half the original population of State A.
**Question 12:**
Graphing \( f(x) = 18(0.8)^x \):
- **Correct Answer**: an exponential graph decreasing from the left and crossing the y axis at 0, 18.
**Question 13:**
For the bacteria decay graph:
- **Correct Answer**: 0 ≤ x ≤ 18, −3.33.
**Question 14:**
In the savings account function:
- **Correct Answer**: The initial amount of money placed in the savings account.
**Question 15:**
Analyzing the expression:
- **Correct Answer**: It is irrational and equal to 4 times the square root of 2. (Note: \( \sqrt{18} = 3\sqrt{2} \), resulting in \( 3\sqrt{2} + \sqrt{2} = 4\sqrt{2} \), which does not simplify neatly, thus is irrational).
Let me know if you need help with anything else!
**Question 1:**
The function \( f(x) = 10(0.8)^x \) represents the amount of pollution left after passing through \( x \) feet of the clay pipe. The number 0.8 represents:
- **Correct Answer**: Every foot of the pipe keeps 80% of the pollutant.
**Question 2:**
Given the profits of the two businesses, we need to determine which function is exponential by observing the profit values:
- **Correct Answer**: f(x) is exponential; f(x) increased more overall than g(x).
**Question 3:**
The function \( f(x) = 46(0.75)^x \) indicates decay because 0.75 signifies a reduction per unit increase in \( x \):
- **Correct Answer**: 75% decay.
**Question 4:**
The function \( f(x) = 16(0.75)^x \) indicates decay since the base is less than 1:
- **Correct Answer**: Exponential decay, 25% decrease. (The decay per time period is 1 - 0.75 = 0.25, or 25%.)
**Question 5:**
The bacteria doubling every minute is expressed as:
- **Correct Answer**: 0 ≤ x ≤ 50 (from 9:05 a.m. to 9:55 a.m., it totals 50 minutes).
**Question 6:**
The balance increase from week to week reflects a non-linear relationship:
- **Correct Answer**: This is a linear function because there is a common difference in the balance between the weeks.
**Question 7:**
To convert weekly growth to daily growth, we need to adjust the base as follows:
- **Correct Answer**: \( f(x) = 86(1.01)^{7x} \); grows approximately at a rate of 1% daily. (Daily rate can be estimated from \( (1.08)^{\frac{1}{7}} - 1 \approx 0.0101 \)).
**Question 8:**
Simplifying the expression involves use of fractional exponents:
- **Correct Answer**: five to the seven sixths power.
**Question 9:**
The function \( y = 14 \cdot (1/2)^x \) indicates decay and the equation of the asymptote:
- **Correct Answer**: Decay; y = 0.
**Question 10:**
For the price increase of 10% per year:
- **Correct Answer**: \( y = 120(1.1)^x \).
**Question 11:**
The initial population comparison:
- **Correct Answer**: The original population of State B was half the original population of State A.
**Question 12:**
Graphing \( f(x) = 18(0.8)^x \):
- **Correct Answer**: an exponential graph decreasing from the left and crossing the y axis at 0, 18.
**Question 13:**
For the bacteria decay graph:
- **Correct Answer**: 0 ≤ x ≤ 18, −3.33.
**Question 14:**
In the savings account function:
- **Correct Answer**: The initial amount of money placed in the savings account.
**Question 15:**
Analyzing the expression:
- **Correct Answer**: It is irrational and equal to 4 times the square root of 2. (Note: \( \sqrt{18} = 3\sqrt{2} \), resulting in \( 3\sqrt{2} + \sqrt{2} = 4\sqrt{2} \), which does not simplify neatly, thus is irrational).
Let me know if you need help with anything else!