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) $$ P=8,000 e^{0.15 t} $$ $$ A(t)=8,000(1+0.05)^{t} $$ $$ P=8,000 e^{0.05 t} $$ $$ A(t)=8,000\left(1+\frac{0.15}{3}\right)^{3 t} $$

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UpStudy AI · Accepted Answer

We start with the formula for continuously compounded interest:

$$
A(t)=P\cdot e^{rt}
$$

Here, $$P = 8000$$ and after $$t = 3$$ years, the amount is $$A(3)=9294.67$$. Thus, we have:

$$
9294.67 = 8000\cdot e^{3r}
$$

Divide both sides by 8000:

$$
e^{3r} = \frac{9294.67}{8000} \approx 1.16183375
$$

Take the natural logarithm of both sides:

$$
3r = \ln(1.16183375)
$$

Using a calculator:

$$
\ln(1.16183375) \approx 0.15
$$

Now, solve for $$r$$:

$$
r = \frac{0.15}{3} = 0.05
$$

Thus, the exponential formula is:

$$
A(t) = 8000\, e^{0.05t}
$$

Among the given options, the correct statement is:

$$
P=8000\;e^{0.05t}
$$
The correct exponential formula is $$ P = 8000 \cdot e^{0.05t} $$.

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