Mr. and Mrs. Thomas hope to send their son to college in thirteen years. How much money should they invest now at an interest rate of $9 \%$ 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
So, Mr. and Mrs. Thomas should invest approximately $2948.49 now.
Step 1 ：The problem is asking for the present value of an investment that will yield $9500 in 13 years with a continuous interest rate of 9%.
Step 2 ：We can use the formula for continuous compounding to find the present value. The formula is \(P = \frac{A}{e^{rt}}\), where P is the present value, A is the future value, r is the interest rate (as a decimal), t is the time in years, and e is the base of the natural logarithm (approximately 2.71828).
Step 3 ：Let's plug in the given values into this formula. We have A = 9500, r = 0.09, and t = 13.
Step 4 ：Calculating the present value, we get \(P = \frac{9500}{e^{0.09 \times 13}}\).
Step 5 ：Computing the above expression, we find that the present value P is approximately 2948.485942022108.
Step 6 ：Rounding to the nearest cent, we get \(\boxed{2948.49}\).
Step 7 ：So, Mr. and Mrs. Thomas should invest approximately $2948.49 now.
