Exponential and Logarithmic Functions
Finding the initial amount in a word problem on continuous compound...
Mr. and Mrs. Phillips hope to send their son to college in fourteen years. How much money should they invest now at an interest rate of $9.5 \%$ per year, compounded continuously, in order to be able to contribute $\$ 7500$ to his education?
Do not round any intermediate computations, and round your answer to the nearest cent.
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Answer
\(\boxed{\text{Final Answer: The initial amount that Mr. and Mrs. Phillips should invest now is approximately \$1983.58.}}\)
Step 1 ：Given values are: final amount \(A = \$7500\), annual interest rate \(r = 9.5\% = 0.095\) in decimal, and time \(t = 14\) years.
Step 2 ：We need to find the initial amount \(P\) that should be invested now.
Step 3 ：We use the formula for continuous compounding, which is \(A = Pe^{rt}\).
Step 4 ：Rearranging the formula to solve for \(P\), we get \(P = \frac{A}{e^{rt}}\).
Step 5 ：Substituting the given values into the formula, we get \(P = \frac{7500}{e^{0.095 \times 14}}\).
Step 6 ：Calculating the above expression, we get \(P \approx \$1983.58\).
Step 7 ：\(\boxed{\text{Final Answer: The initial amount that Mr. and Mrs. Phillips should invest now is approximately \$1983.58.}}\)
