Question

25 POINTS!!! Tommy has $350 of his graduation gift money saved at home, and the amount is modeled by the function h(x) = 350. He reads about a bank that has savings accounts that accrue interest according to the function s(x) = (1.04)x−1. Explain how Tommy can combine the two functions to model the total amount of money he will have in his bank account as interest accrues after he deposits his $350. Justify your reasoning.

Answer 1

To model the total amount of money in Tommy's bank account, the functions h(x) = 350 and s(x) = (1.04)^x−1 are combined into T(x) = 350 + 350 × (1.04)^x−1. This represents the initial savings plus compound interest accrued over x years.

Step-by-step explanation:

Tommy can combine his personal savings function with the bank's interest accrual function to determine the total amount of money in his bank account over time. Since Tommy has $350 saved at home, represented by the function h(x) = 350, and the bank's savings account grows according to the function s(x) = (1.04)x−1, the total money, T(x), after x years can be represented by:

T(x) = 350 + 350 × (1.04)x−1

This equation combines Tommy's initial savings with the growth due to compound interest from the bank. The initial $350 is constant, and the multiplication by s(x) represents the growth of that $350 over x years at a 4% annual interest rate. The usage of (1.04)x−1 stems from the nature of compound interest where the initial amount is multiplied by the growth factor, which includes the initial principal plus the accrued interest after each period.