mr. and mrs. smith hope to send their son to college in twelve years. how much money should they invest now…

mr. and mrs. smith hope to send their son to college in twelve 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 $9500 to his education? do not round any intermediate computations, and round your answer to the nearest cent.

mr. and mrs. smith hope to send their son to college in twelve 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 $9500 to his education? do not round any intermediate computations, and round your answer to the nearest cent.

Answer

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount after time $t$, $P$ is the principal (initial investment), $r$ is the annual interest rate (in decimal), and $t$ is the time in years. We need to find $P$ when $A = 9500$, $r = 0.085$ (since $8.5%=0.085$), and $t = 12$. Rearranging the formula to solve for $P$ gives $P=\frac{A}{e^{rt}}$.

Step2: Substitute the given values into the formula

First, calculate $rt$: $rt=0.085\times12 = 1.02$. Then, calculate $e^{rt}=e^{1.02}$. Using a calculator, $e^{1.02}\approx2.77408$. Now, substitute $A = 9500$ and $e^{rt}\approx2.77408$ into the formula for $P$: $P=\frac{9500}{2.77408}\approx3424.56$.

Answer:

$$3424.56$