5. Find the value of 25,000 invested at 7.5% for 11 years if it is compounded monthly Eqn __ Ans __ 6. Find the value of 25,000 invested for 11 years if it is compounded continuously at 7.5% Eqn __ Ans __

5. To find the value of $25,000 invested at 7.5% for 11 years compounded monthly, we can use the formula for compound interest:<br /><br />\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]<br /><br />where:<br />- \( A \) is the amount of money accumulated after n years, including interest.<br />- \( P \) is the principal amount (the initial amount of money).<br />- \( r \) is the annual interest rate (decimal).<br />- \( n \) is the number of times that interest is compounded per year.<br />- \( t \) is the time the money is invested for in years.<br /><br />Given:<br />- \( P = 25000 \)<br />- \( r = 0.075 \)<br />- \( n = 12 \) (since it is compounded monthly)<br />- \( t = 11 \)<br /><br />Plugging in the values:<br /><br />\[ A = 25000 \left(1 + \frac{0.075}{12}\right)^{12 \times 11} \]<br /><br />\[ A = 25000 \left(1 + 0.00625\right)^{132} \]<br /><br />\[ A = 25000 \left(1.00625\right)^{132} \]<br /><br />\[ A \approx 25000 \times 2.11367 \]<br /><br />\[ A \approx 52683.25 \]<br /><br />So, the value of $25,000 invested at 7.5% for 11 years compounded monthly is approximately $52,683.25.<br /><br />6. To find the value of $25,000 invested for 11 years compounded continuously, we can use the formula for continuous compounding:<br /><br />\[ A = Pe^{rt} \]<br /><br />where:<br />- \( A \) is the amount of money accumulated after n years, including interest.<br />- \( P \) is the principal amount (the initial amount of money).<br />- \( r \) is the annual interest rate (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 />- \( P = 25000 \)<br />- \( r = 0.075 \)<br />- \( t = 11 \)<br /><br />Plugging in the values:<br /><br />\[ A = 25000 \times e^{0.075 \times 11} \]<br /><br />\[ A = 25000 \times e^{0.825} \]<br /><br />\[ A \approx 25000 \times 2.287 \]<br /><br />\[ A \approx 57175 \]<br /><br />So, the value of $25,000 invested for 11 years compounded continuously at 7.5% is approximately $57,175.