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

Ask by Allan Bowers. in the United States
Mar 12,2025

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

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To determine the correct model for Jason's loan growth, we know that the loan amount grows continuously according to the formula \( A(t) = Pe^{rt} \), where \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years. Over three years, Jason started with \( P = 8000 \) and ended up with \( A(3) = 9294.67 \). By setting up the equation \( 9294.67 = 8000 e^{3r} \), we can solve for \( r \) to find the interest rate. The correct model will use the correct growth rate derived from this equation. Now, if we check the statements given, we identify that \( P=8,000 e^{0.05 t} \) uses an interest rate of approximately \( 5\% \) (which seems too low given the final amount), while \( P=8,000 e^{0.15 t} \) uses \( 15\% \) (which is likely the more reasonable rate given the growth in amount over time). In summation, to find the correct option: **\( P=8,000 e^{0.15 t} \) is the correct equation.** In the world of finance, continuous compounding essentially means your money grows on itself, much like a snowball effect—gathering more snow as it rolls down the hill. This compounding also means that interest is calculated on top of previously accumulated interest, which can lead to a much larger amount owed or gained over time compared to simple interest. Don't forget that the world of loans and interest rates can easily get rocky! Always double-check your calculations and understand your loan terms. Lenders often use terms written in a confusing manner—like APR vs. effective interest rates—so take your time to unravel the small print to avoid any financial slip-ups!

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