Amira is going to invest in an account paying an interest rate of 2.4% compounded continuously. How much would Amira need to invest, to the nearest dollar, for the value of the account to reach $320 in 17 years?
Question
Amira is going to invest in an account paying an interest rate of 2.4% compounded continuously. How much would Amira need to invest, to the nearest dollar, for the value of the account to reach $320 in 17 years?
Solution
To solve this problem, we can use the formula for continuous compound interest, which is A = Pe^(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.
We know that A = $320, r = 2.4% or 0.024 (in decimal), and t = 17 years. We want to find P.
Rearranging the formula to solve for P gives us P = A / e^(rt).
Substituting the given values into the formula gives us P = 320 / e^(0.024*17).
Calculating this gives us P = $192.68.
Rounding to the nearest dollar gives us 193 for the value of the account to reach $320 in 17 years.
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