Question
Jason takes out a loan of $8,000.00 from his bank to build a front porch. After three years, not having made any payments toward the loan, he owes $9,294.67. His bank compounds the interest on the loan continuously, and the account has a fixed interest rate. Create an exponential formula to model the growth rate of his loan, rounding the annual interest rate to two decimal places. Which of the following statements is correct?(1 point)
Responses
A(t)=8,000(1+0.05)t
upper A left parenthesis t right parenthesis equals 8,000 left parenthesis 1 plus 0.05 right parenthesis superscript t baseline
P=8,000e0.15t
upper P equals 8,000 e superscript 0.15 5 baseline
A(t)=8,000(1+0.153)3t
upper A left parenthesis t right parenthesis equals 8,000 left parenthesis 1 plus Start Fraction 0.15 over 3 End Fraction right parenthesis superscript 3 t baseline
P=8,000e0.05t
Responses
A(t)=8,000(1+0.05)t
upper A left parenthesis t right parenthesis equals 8,000 left parenthesis 1 plus 0.05 right parenthesis superscript t baseline
P=8,000e0.15t
upper P equals 8,000 e superscript 0.15 5 baseline
A(t)=8,000(1+0.153)3t
upper A left parenthesis t right parenthesis equals 8,000 left parenthesis 1 plus Start Fraction 0.15 over 3 End Fraction right parenthesis superscript 3 t baseline
P=8,000e0.05t
Answers
GPT-4o mini
To model the growth of Jason's loan using continuous compounding, we start with the formula for continuous compounding, which is:
\[
P(t) = P_0 e^{rt}
\]
Where:
- \( P(t) \) is the amount owed at time \( t \),
- \( P_0 \) is the initial principal (the original loan amount),
- \( r \) is the annual interest rate,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
Given:
- \( P_0 = 8000 \)
- \( P(3) = 9294.67 \) (the amount owed after 3 years)
- \( t = 3 \)
We want to find \( r \). Setting up the equation, we get:
\[
9294.67 = 8000 e^{3r}
\]
To isolate \( e^{3r} \):
\[
\frac{9294.67}{8000} = e^{3r}
\]
Calculating the left side:
\[
\frac{9294.67}{8000} \approx 1.16183375
\]
Now take the natural logarithm of both sides:
\[
\ln(1.16183375) = 3r
\]
Calculate \( \ln(1.16183375) \):
\[
\ln(1.16183375) \approx 0.149199
\]
Now, solve for \( r \):
\[
r = \frac{0.149199}{3} \approx 0.049733
\]
To express \( r \) as a percentage, multiply by 100:
\[
r \approx 4.97\% \text{ (approximately 0.05 when rounded to two decimal places)}
\]
Now we can write the formula for the amount owed as:
\[
P(t) = 8000 e^{0.049733t}
\]
Since \( 0.05 \) rounded to two decimal places is close to \( 0.05 \), we can express the final answer using this approximation.
Now, looking at the options given:
1. **A(t) = 8,000(1 + 0.05)^t** (This uses discrete compounding)
2. **P = 8,000e^{0.15t}** (Wrong rate)
3. **A(t) = 8,000(1 + 0.153)^3t** (Wrong rate and wrong method)
4. **P = 8,000e^{0.05t}** (Correct approximation)
The correct answer is:
**P = 8,000e^{0.05t}**.
\[
P(t) = P_0 e^{rt}
\]
Where:
- \( P(t) \) is the amount owed at time \( t \),
- \( P_0 \) is the initial principal (the original loan amount),
- \( r \) is the annual interest rate,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
Given:
- \( P_0 = 8000 \)
- \( P(3) = 9294.67 \) (the amount owed after 3 years)
- \( t = 3 \)
We want to find \( r \). Setting up the equation, we get:
\[
9294.67 = 8000 e^{3r}
\]
To isolate \( e^{3r} \):
\[
\frac{9294.67}{8000} = e^{3r}
\]
Calculating the left side:
\[
\frac{9294.67}{8000} \approx 1.16183375
\]
Now take the natural logarithm of both sides:
\[
\ln(1.16183375) = 3r
\]
Calculate \( \ln(1.16183375) \):
\[
\ln(1.16183375) \approx 0.149199
\]
Now, solve for \( r \):
\[
r = \frac{0.149199}{3} \approx 0.049733
\]
To express \( r \) as a percentage, multiply by 100:
\[
r \approx 4.97\% \text{ (approximately 0.05 when rounded to two decimal places)}
\]
Now we can write the formula for the amount owed as:
\[
P(t) = 8000 e^{0.049733t}
\]
Since \( 0.05 \) rounded to two decimal places is close to \( 0.05 \), we can express the final answer using this approximation.
Now, looking at the options given:
1. **A(t) = 8,000(1 + 0.05)^t** (This uses discrete compounding)
2. **P = 8,000e^{0.15t}** (Wrong rate)
3. **A(t) = 8,000(1 + 0.153)^3t** (Wrong rate and wrong method)
4. **P = 8,000e^{0.05t}** (Correct approximation)
The correct answer is:
**P = 8,000e^{0.05t}**.