Finding the initial amount in a word problem on continuous compound.
How much should be invested now at an interest rate of $6.5 \%$ per year, compounded continuously, to have $\$ 1500$ in four years?
Do not round any intermediate computations, and round your answer to the nearest cent.
Answer
Thus, the initial amount that should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $1500 in four years is \(\boxed{1156.58}\).
Step 1 ：Given that the final amount (A) is $1500, the interest rate (r) is 0.065 (or 6.5%), and the time (t) is 4 years, we need to find the initial amount (P) to be invested.
Step 2 ：We use the formula for continuous compounding, which is A = P * e^(rt).
Step 3 ：We rearrange this formula to solve for P: P = A / e^(rt).
Step 4 ：Substituting the given values into the formula, we get P = 1500 / e^(0.065*4).
Step 5 ：Calculating the above expression, we find that P = 1156.58.
Step 6 ：Thus, the initial amount that should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $1500 in four years is \(\boxed{1156.58}\).
