Question

$190 is invested in an account earning 7.1% interest (APR), compounded quarterly. 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

Value of account after t years is

`A = \boxed{190( 1.0178)^(4t)}`

Percentage of growth per year

CAGR =
`\boxed{7.31\%}`

Explanation:

The generalized equation for the accrued value of an amount P(the principal) deposited at i% annual interest compounded n times a year for t years is

`A = P\left(1 + (r)/(n)\right)^(nt)`

where r = i/100, the interest rate expressed as a decimal

Given P = $190
i = 7.1% ==> r = 7.1/100 = 0.071

Since the interest is compounded quarterly and there are 4 quarters in a year, the interest is compounded 4 times a year so n = 4

`A = 190 \left(1 + (0.071)/(4)\right)^(4t)`

`A = 190\left(1 + 0.01775\right)^(4t)}\\\\A = 190( 1.01775)^(4t)`\

Rounding the annual growth rate to 4 decimal places gives

`A = P(1.0178)^(4t)`

In one year, the investment of $190 will grow to

`A = 190(1.0178)^(4)`

`A = $203.89`

Compound Annual Growth = = 203.89 - 190 = $13.89

Compound Annual Growth Rage(CAGR) = $13.89/$190 = 0.073105

As a percentage, CAGR Percent = 0.073105 x 100 = 7.3105% or rounded to nearest hundredth it would be 7.31%