Question
Which answer best explains the relationship between simple interest and a linear function?(1 point)
Responses
Simple interest earned is the same each year much like a linear function has a graph that is curved.
Simple interest earned is the same each year much like a linear function has a graph that is curved.
Simple interest earned graphed shows a curve, while a linear function is a line.
Simple interest earned graphed shows a curve, while a linear function is a line.
When calculating simple interest, the value can change during different years much like a linear function can have many different points.
When calculating simple interest, the value can change during different years much like a linear function can have many different points.
Simple interest earned is the same each year much like the slope of a linear function is always the same.
Simple interest earned is the same each year much like the slope of a linear function is always the same.
Responses
Simple interest earned is the same each year much like a linear function has a graph that is curved.
Simple interest earned is the same each year much like a linear function has a graph that is curved.
Simple interest earned graphed shows a curve, while a linear function is a line.
Simple interest earned graphed shows a curve, while a linear function is a line.
When calculating simple interest, the value can change during different years much like a linear function can have many different points.
When calculating simple interest, the value can change during different years much like a linear function can have many different points.
Simple interest earned is the same each year much like the slope of a linear function is always the same.
Simple interest earned is the same each year much like the slope of a linear function is always the same.
Answers
GPT-4o mini
The best explanation of the relationship between simple interest and a linear function is:
**Simple interest earned is the same each year much like the slope of a linear function is always the same.**
This is because simple interest is calculated using a constant rate applied to the principal amount, leading to a consistent increase over time, which is characteristic of a linear function with a constant slope.
**Simple interest earned is the same each year much like the slope of a linear function is always the same.**
This is because simple interest is calculated using a constant rate applied to the principal amount, leading to a consistent increase over time, which is characteristic of a linear function with a constant slope.
x
Consider this formula for finding the total amount of money generated by an account receiving compound interest: A=200(1+0.081)1⋅t
. When comparing to y=a(1+r)t
, the general formula for exponential growth, which part of the compound interest formula compares best with the variable r?(1 point)
Responses
0.08
0.08
1
1
1t
1 t
200
. When comparing to y=a(1+r)t
, the general formula for exponential growth, which part of the compound interest formula compares best with the variable r?(1 point)
Responses
0.08
0.08
1
1
1t
1 t
200
GPT-4o mini
In the compound interest formula \( A = 200(1 + 0.081)^{1 \cdot t} \), the part that compares best with the variable \( r \) in the general formula for exponential growth \( y = a(1 + r)^t \) is:
**0.08**
This is because \( r \) represents the growth rate, and in this equation, \( (1 + 0.081) \) shows that the growth rate is \( 0.081 \).
**0.08**
This is because \( r \) represents the growth rate, and in this equation, \( (1 + 0.081) \) shows that the growth rate is \( 0.081 \).
x
A high school student deposits their summer job earnings of $800 into a simple interest savings plan with a 15% rate. How much will the student have earned after 4 years?(1 point)
Responses
$120.00
$120.00
$48,000.00
$48,000.00
$480
$480
$1399.21
$1399.21
Responses
$120.00
$120.00
$48,000.00
$48,000.00
$480
$480
$1399.21
$1399.21
GPT-4o mini
To calculate the amount earned from a simple interest savings plan, you can use the formula:
\[ \text{Interest} = P \times r \times t \]
where:
- \( P \) = principal amount (the initial deposit)
- \( r \) = annual interest rate (in decimal form)
- \( t \) = time (in years)
For this example:
- \( P = 800 \)
- \( r = 0.15 \) (15% expressed as a decimal)
- \( t = 4 \)
Plugging in the values:
\[ \text{Interest} = 800 \times 0.15 \times 4 \]
\[ \text{Interest} = 800 \times 0.15 = 120 \]
\[ \text{Interest} = 120 \times 4 = 480 \]
So the student will have earned **$480** after 4 years.
The correct response is:
**$480**
\[ \text{Interest} = P \times r \times t \]
where:
- \( P \) = principal amount (the initial deposit)
- \( r \) = annual interest rate (in decimal form)
- \( t \) = time (in years)
For this example:
- \( P = 800 \)
- \( r = 0.15 \) (15% expressed as a decimal)
- \( t = 4 \)
Plugging in the values:
\[ \text{Interest} = 800 \times 0.15 \times 4 \]
\[ \text{Interest} = 800 \times 0.15 = 120 \]
\[ \text{Interest} = 120 \times 4 = 480 \]
So the student will have earned **$480** after 4 years.
The correct response is:
**$480**
x
About how much interest will be earned on a principal of $675 deposited into an account paying 3% interest compounded quarterly (4 times per year) for 6 years?(1 point)
Responses
$121.50
$121.50
$130.99
$130.99
$132.58
$132.58
$807.58
Responses
$121.50
$121.50
$130.99
$130.99
$132.58
$132.58
$807.58
GPT-4o mini
To calculate the interest earned on a principal amount with compound interest, you can use the formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}
\]
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = principal amount (the initial deposit).
- \( r \) = annual interest rate (decimal).
- \( n \) = number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested or borrowed.
Given:
- \( P = 675 \)
- \( r = 0.03 \) (3% expressed as a decimal)
- \( n = 4 \) (quarterly)
- \( t = 6 \)
Now substitute these values into the formula:
\[
A = 675 \left(1 + \frac{0.03}{4}\right)^{4 \cdot 6}
\]
Calculating the values step by step:
1. Calculate \( \frac{0.03}{4} = 0.0075 \).
2. Then calculate \( 1 + 0.0075 = 1.0075 \).
3. Calculate \( 4 \cdot 6 = 24 \).
4. Now calculate \( A = 675 \times (1.0075)^{24} \).
Now we can calculate \( (1.0075)^{24} \):
\[
(1.0075)^{24} \approx 1.197427
\]
Now substitute it back into the equation for \( A \):
\[
A \approx 675 \times 1.197427 \approx 807.58
\]
To find the interest earned, subtract the principal from \( A \):
\[
\text{Interest} = A - P \approx 807.58 - 675 = 132.58
\]
So the interest earned is approximately **$132.58**.
The correct response is:
**$132.58**
\[
A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}
\]
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = principal amount (the initial deposit).
- \( r \) = annual interest rate (decimal).
- \( n \) = number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested or borrowed.
Given:
- \( P = 675 \)
- \( r = 0.03 \) (3% expressed as a decimal)
- \( n = 4 \) (quarterly)
- \( t = 6 \)
Now substitute these values into the formula:
\[
A = 675 \left(1 + \frac{0.03}{4}\right)^{4 \cdot 6}
\]
Calculating the values step by step:
1. Calculate \( \frac{0.03}{4} = 0.0075 \).
2. Then calculate \( 1 + 0.0075 = 1.0075 \).
3. Calculate \( 4 \cdot 6 = 24 \).
4. Now calculate \( A = 675 \times (1.0075)^{24} \).
Now we can calculate \( (1.0075)^{24} \):
\[
(1.0075)^{24} \approx 1.197427
\]
Now substitute it back into the equation for \( A \):
\[
A \approx 675 \times 1.197427 \approx 807.58
\]
To find the interest earned, subtract the principal from \( A \):
\[
\text{Interest} = A - P \approx 807.58 - 675 = 132.58
\]
So the interest earned is approximately **$132.58**.
The correct response is:
**$132.58**