EXPONEMTLL AND LOGARTHMIC FUNCTIONS Finding the initial amount in a word problem on continuous... Linda wants to buy a bond that will mature to $\$ 6500$ in eight years. How much should she pay for the bond now if it earns interest at a rate of $3.5 \%$ per year, compounded continuously? Do not round any intermediate computations, and round your answer to the nearest cent.

Step 1 ：The problem is asking for the initial amount Linda should pay for the bond. This is a problem of continuous compounding interest. The formula for continuous compounding is: \(A = P * e^{rt}\) where: \(A\) is the amount of money accumulated after n years, including interest. \(P\) is the principal amount (the initial amount of money) \(r\) is the annual interest rate (in decimal) \(t\) is the time the money is invested for, in years.

Step 2 ：In this case, we know \(A = \$6500\), \(r = 3.5\%\) or \(0.035\) in decimal form, and \(t = 8\) years. We need to solve for \(P\).

Step 3 ：We can rearrange the formula to solve for \(P\): \(P = A / e^{rt}\)

Step 4 ：Substitute the known values into the formula: \(P = 6500 / e^{(0.035*8)}\)

Step 5 ：Solving the equation gives \(P = 4912.594319462216\)

Step 6 ：Rounding to the nearest cent, Linda should pay approximately \$4912.59 for the bond now. This is the initial amount she should pay for the bond.

Step 7 ：So, the final answer is \(\boxed{4912.59}\)