Final answer:
To determine the amount Xavier would have in his account when Suav's investment has tripled in value, we calculate the time it takes for Suav's investment to triple using continuous compounding and then apply that time to calculate Xavier's account balance with annual compound interest.
Step-by-step explanation:
The student's question asks about the final amount in two investment accounts, one with compound interest compounded annually and another with interest compounded continuously. We will first calculate how long it takes for Suav's investment to triple in value at a continuous rate of 6(3/8)%, and then find out how much Xavier's account will have grown in that same period at a rate of 6(3/4)% compounded annually.
Calculating the Time for Suav's Investment to Triple
To find the time (t), we use the formula for continuous compounding, A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (as a decimal), and e is the base of the natural logarithm. We want to solve for t when A is three times the original principal P:
3P = Pe(0.06375)t
3 = e(0.06375)t
ln(3) = 0.06375t
t = ln(3) / 0.06375
Calculating the Amount in Xavier's Account
To calculate the amount in Xavier's account at the same time t, we use the formula for annual compounding interest, A = P(1 + r)n:
A = $6,900(1 + 0.0675)t
We substitute the value of t found from the previous calculation to find the amount to the nearest dollar.