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
P=8,000e0.05t
upper P equals 8,000 e superscript 0.05 t 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
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
Responses
P=8,000e0.05t
upper P equals 8,000 e superscript 0.05 t 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
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
Answers
GPT-4o mini
To create an exponential formula for the loan that Jason has taken, we know that the continuous compounding formula is given by:
\[ P(t) = P_0 e^{rt} \]
where:
- \( P(t) \) is the amount owed after time \( t \),
- \( P_0 \) is the initial principal (the amount of the loan),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- \( P_0 = 8000 \)
- After 3 years, \( P(3) = 9294.67 \)
Using the formula, we can set up the following equation to solve for \( r \):
\[ 9294.67 = 8000 e^{3r} \]
Dividing both sides by 8000:
\[ \frac{9294.67}{8000} = e^{3r} \]
Calculating \( \frac{9294.67}{8000} \):
\[ \frac{9294.67}{8000} = 1.16183375 \]
Now we take the natural logarithm of both sides:
\[ \ln(1.16183375) = 3r \]
Calculating \( \ln(1.16183375) \):
\[ r = \frac{\ln(1.16183375)}{3} \approx \frac{0.149}\3 \approx 0.0497 \]
Thus, the annual interest rate \( r \) is approximately 0.05 (or 5% when expressed as a percentage).
Now, we can write the model for the loan:
\[ P(t) = 8000 e^{0.0497t} \]
Rounded, we can write it using \( r \approx 0.05 \):
\[ P(t) = 8000 e^{0.05t} \]
Among the given options, the statement that matches with our derived model is:
**P = 8,000e^{0.05t}**.
Thus, the correct response is:
**P=8,000e^{0.05t}**.
\[ P(t) = P_0 e^{rt} \]
where:
- \( P(t) \) is the amount owed after time \( t \),
- \( P_0 \) is the initial principal (the amount of the loan),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- \( P_0 = 8000 \)
- After 3 years, \( P(3) = 9294.67 \)
Using the formula, we can set up the following equation to solve for \( r \):
\[ 9294.67 = 8000 e^{3r} \]
Dividing both sides by 8000:
\[ \frac{9294.67}{8000} = e^{3r} \]
Calculating \( \frac{9294.67}{8000} \):
\[ \frac{9294.67}{8000} = 1.16183375 \]
Now we take the natural logarithm of both sides:
\[ \ln(1.16183375) = 3r \]
Calculating \( \ln(1.16183375) \):
\[ r = \frac{\ln(1.16183375)}{3} \approx \frac{0.149}\3 \approx 0.0497 \]
Thus, the annual interest rate \( r \) is approximately 0.05 (or 5% when expressed as a percentage).
Now, we can write the model for the loan:
\[ P(t) = 8000 e^{0.0497t} \]
Rounded, we can write it using \( r \approx 0.05 \):
\[ P(t) = 8000 e^{0.05t} \]
Among the given options, the statement that matches with our derived model is:
**P = 8,000e^{0.05t}**.
Thus, the correct response is:
**P=8,000e^{0.05t}**.