Problem 93 - Precalculus with Limits

Question

You are investing \(P\) dollars at an annual interest rate of \(r,\) compounded continuously, for \(t\) years. Which of the following would result in the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years.

Step-by-Step Solution

Answer

Doubling the interest rate or the time period results in a larger increase in the final investment amount compared to just doubling the initial investment. However, without specific values for P, r, and t, it cannot be conclusively told whether doubling the interest rate or the time period results in a higher value.

Step 1: Double the Initial Investment

To explore the effect of doubling the initial investment, substitute \( 2P \) in place of \( P \) in the formula: \( A = 2Pe^{rt} \). This results in doubling the final amount A. Thus, doubling the initial investment doubles the final amount.

Step 2: Double the Interest Rate

To analyze the effect of doubling the interest rate, substitute \( 2r \) in place of \( r \) in the formula: \( A = Pe^{2rt} \). This doesn't result in a simple doubling of the final amount. Since e is an exponential function, doubling the exponent increases the final amount more than just two times.

Step 3: Double the Time Period

Finally, to gauge the effect of doubling the time period, substitute \( 2t \) in place of \( t \) in the formula: \( A = Pe^{r2t} \). Since t is in the exponent, this also doesn't simply double the final amount. As with doubling the interest rate, the exponential function causes the final amount to increase more than two times.