Yolanda wants to buy a bond that will mature to 5000 in six years . How much should she pay for the bond now if it earns interest at a rate of 2.5% per year,compounded continuously? Do not round any intermediate computations , and round your answer to the nearest cent. square

To solve this problem, we need to use the formula for continuous compounding interest:<br /><br />\[ A = P \cdot e^{rt} \]<br /><br />Where:<br />- \( A \) is the amount of money accumulated after time \( t \)<br />- \( P \) is the principal amount (the initial amount of money)<br />- \( r \) is the annual interest rate (expressed as a decimal)<br />- \( t \) is the time the money is invested for (in years)<br />- \( e \) is the base of the natural logarithm (approximately equal to 2.71828)<br /><br />Given:<br />- \( A = \$5000 \)<br />- \( r = 2.5\% = 0.025 \)<br />- \( t = 6 \) years<br /><br />We need to find \( P \), the amount Yolanda should pay for the bond now.<br /><br />Rearranging the formula to solve for \( P \):<br /><br />\[ P = \frac{A}{e^{rt}} \]<br /><br />Substituting the given values:<br /><br />\[ P = \frac{5000}{e^{0.025 \cdot 6}} \]<br /><br />First, calculate the exponent:<br /><br />\[ 0.025 \cdot 6 = 0.15 \]<br /><br />Next, calculate \( e^{0.15} \):<br /><br />\[ e^{0.15} \approx 1.1618 \]<br /><br />Now, divide value by this amount:<br /><br />\[ P = \frac{5000}{1.1618} \approx 4304.63 \]<br /><br />Therefore, Yolanda should pay approximately \$4304.63 for the bond now.