It takes 10 years for a $100 monthly bank deposit to amount to $15,528.23 when invested at 5% interest compounded monthly. Round to the nearest place value, then express the amount $15,528.23 in the form of a×10n%0D%0A%0D%0A×%0D%0A10%0D%0A%0D%0A where a%0D%0A%0D%0A is single digit.(1 point
To find the interest rate in the form of a×10^n%, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment ($15,528.23)
P = the monthly deposit ($100)
r = the annual interest rate (unknown, to be calculated)
n = the number of times interest is compounded per year (12 for monthly compounding)
t = the number of years the money is invested (10)
Plugging in the given values, we get:
15,528.23 = 100(1 + r/12)^(12*10)
Divide both sides by 100:
155.2823 = (1 + r/12)^120
Take the 120th root of both sides:
(1 + r/12) = 120th root of 155.2823
Now, subtract 1 from both sides:
r/12 = 120th root of 155.2823 - 1
Multiply both sides by 12:
r = 12 * ((120th root of 155.2823) - 1)
Using a calculator, we find that the interest rate (r) is approximately 0.3903 (rounded to four decimal places). Therefore, the interest rate in the form of a×10^n% is 3.903% (rounded to three decimal places).
So, the amount $15,528.23 can be expressed as 3.903×10^3%.