Question

$9,000 is invested in an account earning 8% interest (APR), compounded daily.

Write a function showing the value of the account after t years, where the annual
growth rate can be found from a constant in the function. Round all coefficients in
the function to four decimal places. Also, determine the percentage of growth per
year (APY), to the nearest hundredth of a percent.

Answer 1

The formula for calculating the value of an account with compound interest is:

A = P(1 + r/n)^(nt)

where:

A = the final amount

P = the principal (starting amount)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period in years

For this problem, we have:

P = $9,000

r = 8% = 0.08 (APR)

n = 365 (compounded daily)

t = number of years

So the function for the value of the account after t years is:

f(t) = 9000(1 + 0.08/365)^(365t)

Simplifying and rounding to four decimal places:

f(t) = 9000(1.000219178)^t

f(t) = 9000(1.0002)^t

To determine the annual percentage yield (APY), we can use the formula:

APY = (1 + r/n)^n - 1

where r = 0.08 (APR) and n = 365 (compounded daily)

APY = (1 + 0.08/365)^365 - 1

APY = 0.0833 or 8.33%

So the account is growing at an annual rate of 8.33%.