Question

in 1887, a person sold a house to a lady for $30. if the lady had put the $30 into a bank account paying 6% interest, how much would the investment have been worth in the year 2012 if interest were compounded in the following ways?

1. monthly

b. continuously

1. if compounded monthly, the investment would be worth $ 53,237 in 2012. (round to the nearest dollar as needed.)

b. if compounded continuously, the investment would be worth $\square$ in 2012. (round to the nearest dollar as needed.)

Explanation:

Step1: Determine the time period

First, we find the number of years from 1887 to 2012. So, \( 2012 - 1887 = 125 \) years. The principal amount \( P=\$30 \), the annual interest rate \( r = 6\%=0.06 \), and for continuous compounding, the formula is \( A = Pe^{rt} \).

Step2: Apply the continuous compounding formula

Substitute the values into the formula \( A = Pe^{rt} \). Here, \( P = 30 \), \( r=0.06 \), and \( t = 125 \). So we calculate \( A=30\times e^{0.06\times125} \). First, calculate the exponent: \( 0.06\times125 = 7.5 \). Then, \( e^{7.5}\approx1808.047 \). Now, multiply by 30: \( 30\times1808.047\approx54241.41 \). Rounding to the nearest dollar, we get \( 54241 \).

Answer:

\( 54241 \)