Final answer:
To find how much Carter would have when Savannah's account doubles, first calculate the time it takes for Savannah's investment to double using the compound interest formula. Then, use that time in the continuous compounding formula to find Carter's account value.
Step-by-step explanation:
The question asks to calculate the future value of Carter's investment when Savannah's investment has doubled using two different types of compound interest calculations. Carter's investment grows with continuous compounding, while Savannah's grows with quarterly compounding. To answer this, we need to find how long it takes for Savannah's investment to double, then apply that time to Carter's investment.
Calculating Savannah's Investment Doubling Time
Since Savannah's rate is compounded quarterly, we use the formula for compound interest given by A = P(1 + r/n)nt where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times the interest is compounded per year, and t is the time the money is invested for, in years. Doubling the investment means A = 2P. Assuming P = $70,000 and r = 1 3/8 % or 0.01375 (as a decimal), compounded quarterly (n=4), we set up the equation and solve for t:
70000(1 + 0.01375 / 4)4t = 140000
Calculating Carter's Investment Future Value
For continuous compounding, the formula is A = Pert, where e is the base of the natural logarithm. Since we know the time t from Savannah's calculation, we can apply this to Carter's investment. Using Carter's principal of $70,000 and a rate of 1 1/8 % or 0.01125, we calculate Carter's account value at the same time Savannah's has doubled.