Question
Karen purchased a car for $40,000 by taking out a loan that will take 7 years to pay off. Given the car depreciates in value by 11% each year, the situation can be modeled by the equation v(t)=40,000(0.89)t
, where v is the value of the car after t years. If Karen wants to know how much the car will be worth in 7 years, which of the following options would be an appropriate range?
Option #1: 0<t≤40,000
Option #2: 0<v≤40,000
Option #3: 0<t≤7
Option #4: 0<v≤7
, where v is the value of the car after t years. If Karen wants to know how much the car will be worth in 7 years, which of the following options would be an appropriate range?
Option #1: 0<t≤40,000
Option #2: 0<v≤40,000
Option #3: 0<t≤7
Option #4: 0<v≤7
Answers
GPT-4o mini
To determine the appropriate range for the value of the car after 7 years, we need to focus on what Karen is interested in: the value of the car, \(v(t)\), after 7 years.
The equation given is:
\[ v(t) = 40,000(0.89)^t \]
We want to find \(v(7)\):
\[ v(7) = 40,000(0.89)^7 \]
Calculating \(0.89^7\):
\[ 0.89^7 \approx 0.513 \] (approximately)
Now, calculating \(v(7)\):
\[ v(7) \approx 40,000 \times 0.513 \]
\[ v(7) \approx 20,520 \]
Now that we have the approximate value of the car after 7 years, we can assess the options provided:
1. **Option #1: 0<t≤40,000** - This option represents a range for \(t\) (time), not for \(v\) (value).
2. **Option #2: 0<v≤40,000** - This is a reasonable range since the value of the car starts at $40,000, and $20,520 (the value after 7 years) falls within this range.
3. **Option #3: 0<t≤7** - This option is also for \(t\), not for \(v\).
4. **Option #4: 0<v≤7** - This option suggests that the car's value is between 0 and 7, which is incorrect based on our calculations.
The most appropriate range to describe the value of the car after 7 years is:
**Option #2: 0<v≤40,000**.
The equation given is:
\[ v(t) = 40,000(0.89)^t \]
We want to find \(v(7)\):
\[ v(7) = 40,000(0.89)^7 \]
Calculating \(0.89^7\):
\[ 0.89^7 \approx 0.513 \] (approximately)
Now, calculating \(v(7)\):
\[ v(7) \approx 40,000 \times 0.513 \]
\[ v(7) \approx 20,520 \]
Now that we have the approximate value of the car after 7 years, we can assess the options provided:
1. **Option #1: 0<t≤40,000** - This option represents a range for \(t\) (time), not for \(v\) (value).
2. **Option #2: 0<v≤40,000** - This is a reasonable range since the value of the car starts at $40,000, and $20,520 (the value after 7 years) falls within this range.
3. **Option #3: 0<t≤7** - This option is also for \(t\), not for \(v\).
4. **Option #4: 0<v≤7** - This option suggests that the car's value is between 0 and 7, which is incorrect based on our calculations.
The most appropriate range to describe the value of the car after 7 years is:
**Option #2: 0<v≤40,000**.
Use the table to answer the question.
x f(x)
0 3
1 3.6
2 4.32
3 5.18
4 ?
5 7.46
Find the missing value in the table for the exponential function f(x)=3(1.2)x
. Round your answer to two decimal places.
(1 point)
f(4)=
x f(x)
0 3
1 3.6
2 4.32
3 5.18
4 ?
5 7.46
Find the missing value in the table for the exponential function f(x)=3(1.2)x
. Round your answer to two decimal places.
(1 point)
f(4)=
GPT-4o mini
To find the missing value \( f(4) \) using the exponential function given by the equation:
\[
f(x) = 3(1.2)^x
\]
we can substitute \( x = 4 \) into the equation:
\[
f(4) = 3(1.2)^4
\]
Now, we need to compute \( (1.2)^4 \):
\[
(1.2)^4 = 1.2 \times 1.2 \times 1.2 \times 1.2
\]
Calculating step by step:
\[
(1.2)^2 = 1.44
\]
\[
(1.2)^3 = 1.44 \times 1.2 = 1.728
\]
\[
(1.2)^4 = 1.728 \times 1.2 \approx 2.0736
\]
Now, substituting back into the equation for \( f(4) \):
\[
f(4) = 3 \times 2.0736 \approx 6.2208
\]
Rounding to two decimal places:
\[
f(4) \approx 6.22
\]
Thus, the missing value in the table is:
\[
f(4) = 6.22
\]
\[
f(x) = 3(1.2)^x
\]
we can substitute \( x = 4 \) into the equation:
\[
f(4) = 3(1.2)^4
\]
Now, we need to compute \( (1.2)^4 \):
\[
(1.2)^4 = 1.2 \times 1.2 \times 1.2 \times 1.2
\]
Calculating step by step:
\[
(1.2)^2 = 1.44
\]
\[
(1.2)^3 = 1.44 \times 1.2 = 1.728
\]
\[
(1.2)^4 = 1.728 \times 1.2 \approx 2.0736
\]
Now, substituting back into the equation for \( f(4) \):
\[
f(4) = 3 \times 2.0736 \approx 6.2208
\]
Rounding to two decimal places:
\[
f(4) \approx 6.22
\]
Thus, the missing value in the table is:
\[
f(4) = 6.22
\]
Use the table to answer the question.
x 320
0
1
2
3
4
5
Cason is scheduled to get his wisdom teeth removed. The oral surgeon gives him a local anesthetic to numb his mouth before surgery. Cason is given 320 milligrams of the anesthetic, which metabolizes at a rate of 25% per hour. The situation can be modeled using the exponential equation f(x)=320(0.75)x
. Finish the chart to determine which graph displays the correct plotted points for the situation.
Option #1:
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 7 in increments of 1. The y-axis for milligrams ranges from 0 to 350 in increments of 50.
Option #2:
A curve with an arrow on the top is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 350 in increments of 50. The y-axis for milligrams ranges from 0 to 7 in increments of 1.
Option #3:
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 7 in increments of 1. The y-axis for milligrams ranges from 0 to 350 in increments of 50.
(1 point)
The graph with the correct plotted points is Option #
x 320
0
1
2
3
4
5
Cason is scheduled to get his wisdom teeth removed. The oral surgeon gives him a local anesthetic to numb his mouth before surgery. Cason is given 320 milligrams of the anesthetic, which metabolizes at a rate of 25% per hour. The situation can be modeled using the exponential equation f(x)=320(0.75)x
. Finish the chart to determine which graph displays the correct plotted points for the situation.
Option #1:
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 7 in increments of 1. The y-axis for milligrams ranges from 0 to 350 in increments of 50.
Option #2:
A curve with an arrow on the top is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 350 in increments of 50. The y-axis for milligrams ranges from 0 to 7 in increments of 1.
Option #3:
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for hours ranges from 0 to 7 in increments of 1. The y-axis for milligrams ranges from 0 to 350 in increments of 50.
(1 point)
The graph with the correct plotted points is Option #
GPT-4o mini
To determine which graph displays the correct plotted points for the situation, we first need to calculate the values of \( f(x) \) using the exponential equation:
\[
f(x) = 320(0.75)^x
\]
We will calculate the values for \( x = 0, 1, 2, 3, 4, 5 \):
1. **For \( x = 0 \)**:
\[
f(0) = 320(0.75)^0 = 320 \times 1 = 320
\]
2. **For \( x = 1 \)**:
\[
f(1) = 320(0.75)^1 = 320 \times 0.75 = 240
\]
3. **For \( x = 2 \)**:
\[
f(2) = 320(0.75)^2 = 320 \times 0.5625 = 180
\]
4. **For \( x = 3 \)**:
\[
f(3) = 320(0.75)^3 = 320 \times 0.421875 \approx 135
\]
5. **For \( x = 4 \)**:
\[
f(4) = 320(0.75)^4 = 320 \times 0.3164 \approx 101.25
\]
6. **For \( x = 5 \)**:
\[
f(5) = 320(0.75)^5 = 320 \times 0.2373 \approx 75.36
\]
Now we summarize the computed values:
| \( x \) | \( f(x) \) |
|---------|-----------|
| 0 | 320 |
| 1 | 240 |
| 2 | 180 |
| 3 | 135 |
| 4 | 101.25 |
| 5 | 75.36 |
### Options Review
- **Option #1**: Describes a decreasing curve on a coordinate plane where the x-axis ranges from 0 to 7 (hours) and the y-axis from 0 to 350 (milligrams). This matches our calculated values for \( f(x) \), being a decreasing function.
- **Option #2**: Describes a curve with the y-axis labeled milligrams going from 0 to 350 and the x-axis going from 0 to 7 in increments of 1. This is incorrect since the axes are swapped and does not accurately represent the function.
- **Option #3**: Same as Option #1, describing a decreasing curve plotted with an x-axis for hours from 0 to 7 and a y-axis for milligrams from 0 to 350.
### Conclusion
Both Option #1 and Option #3 describe the function correctly, but the actual correct option number is typically represented as **Option #1**. Thus, the answer is:
**Option #1**.
\[
f(x) = 320(0.75)^x
\]
We will calculate the values for \( x = 0, 1, 2, 3, 4, 5 \):
1. **For \( x = 0 \)**:
\[
f(0) = 320(0.75)^0 = 320 \times 1 = 320
\]
2. **For \( x = 1 \)**:
\[
f(1) = 320(0.75)^1 = 320 \times 0.75 = 240
\]
3. **For \( x = 2 \)**:
\[
f(2) = 320(0.75)^2 = 320 \times 0.5625 = 180
\]
4. **For \( x = 3 \)**:
\[
f(3) = 320(0.75)^3 = 320 \times 0.421875 \approx 135
\]
5. **For \( x = 4 \)**:
\[
f(4) = 320(0.75)^4 = 320 \times 0.3164 \approx 101.25
\]
6. **For \( x = 5 \)**:
\[
f(5) = 320(0.75)^5 = 320 \times 0.2373 \approx 75.36
\]
Now we summarize the computed values:
| \( x \) | \( f(x) \) |
|---------|-----------|
| 0 | 320 |
| 1 | 240 |
| 2 | 180 |
| 3 | 135 |
| 4 | 101.25 |
| 5 | 75.36 |
### Options Review
- **Option #1**: Describes a decreasing curve on a coordinate plane where the x-axis ranges from 0 to 7 (hours) and the y-axis from 0 to 350 (milligrams). This matches our calculated values for \( f(x) \), being a decreasing function.
- **Option #2**: Describes a curve with the y-axis labeled milligrams going from 0 to 350 and the x-axis going from 0 to 7 in increments of 1. This is incorrect since the axes are swapped and does not accurately represent the function.
- **Option #3**: Same as Option #1, describing a decreasing curve plotted with an x-axis for hours from 0 to 7 and a y-axis for milligrams from 0 to 350.
### Conclusion
Both Option #1 and Option #3 describe the function correctly, but the actual correct option number is typically represented as **Option #1**. Thus, the answer is:
**Option #1**.
Use the image to answer the question.
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.
Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t
, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?
Option #1: As t increases, a increases slowly at first and then quickly.
Option #2: The y-intercept of the function is (0,500)
.
Option #3: The range includes all real numbers.
(1 point)
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.
Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t
, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?
Option #1: As t increases, a increases slowly at first and then quickly.
Option #2: The y-intercept of the function is (0,500)
.
Option #3: The range includes all real numbers.
(1 point)
GPT-4o mini
To analyze the statements regarding the graph of the function given by the equation:
\[
a(t) = 500(0.8)^t
\]
we can evaluate each option:
### Option Analysis
1. **Option #1: As t increases, a increases slowly at first and then quickly.**
- This statement is **false**. The function \( a(t) \) is an exponential decay function. As \( t \) increases, \( a(t) \) actually decreases, starting from 500 mg and approaching 0, rather than increasing.
2. **Option #2: The y-intercept of the function is (0, 500).**
- To find the y-intercept, we evaluate \( a(0) \):
\[
a(0) = 500(0.8)^0 = 500 \times 1 = 500
\]
- This statement is **true**. The y-intercept is indeed (0, 500).
3. **Option #3: The range includes all real numbers.**
- The function \( a(t) \) represents the amount of medicine in the bloodstream and is modeled by an exponential decay that approaches 0 but never actually becomes negative. Thus, the range of \( a(t) \) is \( (0, 500] \), not all real numbers. Therefore, this statement is **false**.
### Conclusion
The only true statement about the graph of the function is:
**Option #2: The y-intercept of the function is (0, 500).**
\[
a(t) = 500(0.8)^t
\]
we can evaluate each option:
### Option Analysis
1. **Option #1: As t increases, a increases slowly at first and then quickly.**
- This statement is **false**. The function \( a(t) \) is an exponential decay function. As \( t \) increases, \( a(t) \) actually decreases, starting from 500 mg and approaching 0, rather than increasing.
2. **Option #2: The y-intercept of the function is (0, 500).**
- To find the y-intercept, we evaluate \( a(0) \):
\[
a(0) = 500(0.8)^0 = 500 \times 1 = 500
\]
- This statement is **true**. The y-intercept is indeed (0, 500).
3. **Option #3: The range includes all real numbers.**
- The function \( a(t) \) represents the amount of medicine in the bloodstream and is modeled by an exponential decay that approaches 0 but never actually becomes negative. Thus, the range of \( a(t) \) is \( (0, 500] \), not all real numbers. Therefore, this statement is **false**.
### Conclusion
The only true statement about the graph of the function is:
**Option #2: The y-intercept of the function is (0, 500).**
Use the image to answer the question.
An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The curve begins at the point left parenthesis 0 comma 2,000 right parenthesis.
Nate’s parents invested $2,000 in a savings account when he was born. If the account has a growth rate of 8% per year, the situation can be modeled by the equation a(t)=2,000(1.08)t
, where a is the amount in the investment after t years. Which one of the following statements is true about the graph of the function?
Option #1: As t increases, a increases slowly at first and then quickly.
Option #2: The y-intercept of the function is (0,8)
.
Option #3: The range includes all real numbers.
(1 point)
Option #
is the true statement.
An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The curve begins at the point left parenthesis 0 comma 2,000 right parenthesis.
Nate’s parents invested $2,000 in a savings account when he was born. If the account has a growth rate of 8% per year, the situation can be modeled by the equation a(t)=2,000(1.08)t
, where a is the amount in the investment after t years. Which one of the following statements is true about the graph of the function?
Option #1: As t increases, a increases slowly at first and then quickly.
Option #2: The y-intercept of the function is (0,8)
.
Option #3: The range includes all real numbers.
(1 point)
Option #
is the true statement.
GPT-4o mini
To analyze the given statements regarding the function modeled by the equation:
\[
a(t) = 2000(1.08)^t
\]
we need to evaluate each option based on the behavior of the function.
### Option Analysis
1. **Option #1: As t increases, a increases slowly at first and then quickly.**
- This statement is **true**. The function \( a(t) \) is an exponential growth function. At first, the growth is slow, but as time increases, the growth accelerates rapidly due to the nature of exponential functions.
2. **Option #2: The y-intercept of the function is (0, 8).**
- To find the y-intercept, we evaluate \( a(0) \):
\[
a(0) = 2000(1.08)^0 = 2000 \times 1 = 2000
\]
- This statement is **false**. The correct y-intercept is (0, 2000), not (0, 8).
3. **Option #3: The range includes all real numbers.**
- The function \( a(t) \) represents the amount of money in the investment and will always be positive and increasing as \( t \) increases. Therefore, the range is \( (2000, \infty) \), not all real numbers. This statement is **false**.
### Conclusion
The only true statement about the graph of the function is:
**Option #1: As t increases, a increases slowly at first and then quickly.**
Thus the answer is:
**Option #1**.
\[
a(t) = 2000(1.08)^t
\]
we need to evaluate each option based on the behavior of the function.
### Option Analysis
1. **Option #1: As t increases, a increases slowly at first and then quickly.**
- This statement is **true**. The function \( a(t) \) is an exponential growth function. At first, the growth is slow, but as time increases, the growth accelerates rapidly due to the nature of exponential functions.
2. **Option #2: The y-intercept of the function is (0, 8).**
- To find the y-intercept, we evaluate \( a(0) \):
\[
a(0) = 2000(1.08)^0 = 2000 \times 1 = 2000
\]
- This statement is **false**. The correct y-intercept is (0, 2000), not (0, 8).
3. **Option #3: The range includes all real numbers.**
- The function \( a(t) \) represents the amount of money in the investment and will always be positive and increasing as \( t \) increases. Therefore, the range is \( (2000, \infty) \), not all real numbers. This statement is **false**.
### Conclusion
The only true statement about the graph of the function is:
**Option #1: As t increases, a increases slowly at first and then quickly.**
Thus the answer is:
**Option #1**.
Annabel wants to make banana bread for the bake sale. She went to the market to buy bananas, and she noticed fruit flies after she returned home with her produce. The number of fruit flies grows at an exponential rate modeled by the equation f(t)=8(2)t
, where f is the number of fruit flies after t days. Define an appropriate domain for the problem if Annabel bakes the banana bread 5 days after returning from the market.(1 point)
Responses
0≤t≤256
0 less than or equal to t less than or equal to 256
0≤f≤256
0 less than or equal to f less than or equal to 256
0≤f≤5
0 less than or equal to f less than or equal to 5
0≤t≤5
, where f is the number of fruit flies after t days. Define an appropriate domain for the problem if Annabel bakes the banana bread 5 days after returning from the market.(1 point)
Responses
0≤t≤256
0 less than or equal to t less than or equal to 256
0≤f≤256
0 less than or equal to f less than or equal to 256
0≤f≤5
0 less than or equal to f less than or equal to 5
0≤t≤5
GPT-4o mini
To define an appropriate domain for the problem, we need to understand what the variable \( t \) represents in the given context. In this case, \( t \) represents the number of days after Annabel returns home with her bananas.
The problem states that Annabel bakes the banana bread 5 days after returning from the market. Therefore, the possible values for \( t \) must be between 0 (the day she returns home) and 5 (the day she bakes the banana bread).
This leads us to the conclusion that the correct domain for \( t \) is:
\[
0 \leq t \leq 5
\]
Therefore, the correct choice for defining the appropriate domain is:
**Option:** **0 ≤ t ≤ 5.**
The problem states that Annabel bakes the banana bread 5 days after returning from the market. Therefore, the possible values for \( t \) must be between 0 (the day she returns home) and 5 (the day she bakes the banana bread).
This leads us to the conclusion that the correct domain for \( t \) is:
\[
0 \leq t \leq 5
\]
Therefore, the correct choice for defining the appropriate domain is:
**Option:** **0 ≤ t ≤ 5.**