Question

Maya deposits $900 into an account compounded continuously, that follows the exponential growth model. The initial amount of money in this account doubles every 8.5 years. To the nearest dollar and cents, how much money will Maya have in her account after 5 years? For the rate of growth of this account, round your answer to 3 decimal places.

Answer 1

`{e}^(8.5r) = 2`

`(17)/(2) r = ln(2)`

`r = (2 ln(2) )/(17) = .082`

`900 {e}^{ (2 ln(2) )/(17) * 5 } = 900 {e}^{ (10 ln(2) )/(17) } = 1353.07`

Maya will have $1,353.07 in this account after 5 years.

Answer 2

Using the continuous compounding formula and the rate derived from the doubling time, Maya will have approximately $1317.56 in her account after 5 years, with an annual interest rate of about 8.2%.

Step-by-step explanation:

The student asks how much money will be in an account after 5 years if it's initially deposited with $900 and compounded continuously, with the amount doubling every 8.5 years. To solve this, we'll use the formula for continuous compounding, A = Pert, where P is the principal amount, e is the base of the natural logarithm, r is the annual interest rate, and t is the time in years. Since the account doubles every 8.5 years, we can find the rate using the doubling time formula r = ln(2)/T, where T is the doubling time.

Firstly, calculate the rate:

r = ln(2)/8.5
r ≈ 0.082

Now, we'll use this rate to find the future value after 5 years:

A = 900e(0.082)(5)
A ≈ $900e0.41
A ≈ $1317.56

Therefore, to the nearest dollar and cents, Maya will have $1317.56 in her account after 5 years, with a growth rate of approximately 0.082 or 8.2% (rounded to three decimal places).