mr. and mrs. morgan hope to send their son to college in fourteen years. how much money should they invest…

mr. and mrs. morgan hope to send their son to college in fourteen years. how much money should they invest now at an interest rate of 8.5% per year, compounded continuously, in order to be able to contribute $9000 to his education? do not round any intermediate computations, and round your answer to the nearest cent.

mr. and mrs. morgan hope to send their son to college in fourteen years. how much money should they invest now at an interest rate of 8.5% per year, compounded continuously, in order to be able to contribute $9000 to his education? do not round any intermediate computations, and round your answer to the nearest cent.

Answer

Explanation:

Step1: Recall continuous - compounding formula

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount (initial investment), $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. We need to solve for $P$, so we can rewrite the formula as $P=\frac{A}{e^{rt}}$.

Step2: Convert the interest rate and identify values

The annual interest rate $r = 8.5%=0.085$, the time $t = 14$ years, and the final amount $A = 9000$.

Step3: Substitute values into the formula

$P=\frac{9000}{e^{0.085\times14}}$. First, calculate the exponent: $0.085\times14 = 1.19$. Then, find $e^{1.19}$. Using a calculator, $e^{1.19}\approx3.287037$. So, $P=\frac{9000}{3.287037}\approx2737.90$.

Answer:

$2737.90$