Sophia is saving money and plans on making monthly contributions into an account earning an annual interest rate of 6.3% compounded monthly. If Sophia would like to end up with $65,000 after 11 years, how much does she need to contribute to the account every month, to the nearest dollar? Use the following formula to determine your answer. A=d(((1+i)^(n)-1)/(i)) A= the future value of the account after n periods d= the amount invested at the end of each period i= the interest rate per period n= the number of periods Answer: ◻

Question

Sophia is saving money and plans on making monthly contributions into an account earning an annual interest rate of 6.3% compounded monthly. If Sophia would like to end up with $65,000 after 11 years, how much does she need to contribute to the account every month, to the nearest dollar? Use the following formula to determine your answer.  A=d(((1+i)^(n)-1)/(i))  A= the future value of the account after n periods  d= the amount invested at the end of each period  i= the interest rate per period  n= the number of periods Answer: ◻

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Identify Given Values: Identify the given values from the problem. We are given: Future value of the account, A = $65,000 Annual interest rate, r = 6.3% or 0.063 (as a decimal) Number of years, t = 11 The interest rate per period (monthly), i = r/12 The number of periods, n = t * 12 (since the contributions are monthly)Convert Annual Interest Rate: Convert the annual interest rate to the monthly interest rate. i = r/12 i = 0.063/12 i = 0.00525Calculate Total Periods: Calculate the total number of periods. n = t * 12 n = 11 * 12 n = 132Find Monthly Contribution: Use the formula to find the monthly contribution, d. A = d * (((1 + i)^(n) - 1) / i) $65,000 = d * (((1 + 0.00525)^(132) - 1) / 0.00525)Calculate Value Inside Parentheses: Calculate the value inside the parentheses. (1 + i)^(n) - 1 (1 + 0.00525)^(132) - 1 (1.00525)^(132) - 1Calculate Value of Expression: Calculate the value of the expression (1.00525)^(132) - 1. This requires a calculator or a computer to compute. (1.00525)^(132) - 1 ≈ 0.9994Substitute Calculated Value: Substitute the calculated value back into the formula. $65,000 = d * (0.9994 / 0.00525)Solve for Monthly Contribution: Solve for d, the monthly contribution. d = $65,000 / (0.9994 / 0.00525) d ≈ $65,000 / 0.1902 d ≈ $341.75Round Monthly Contribution: Round the monthly contribution to the nearest dollar. d ≈ $342

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