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[tex]\(\$190\)[/tex] is invested in an account earning [tex]\(2\%\)[/tex] interest (APR), compounded daily. Write a function showing the value of the account after [tex]\(t\)[/tex] 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.
Alright, let's break this down step by step to understand how the formula for compound interest works and determine the percentage of growth per year (APY).
### Compound Interest Formula The general formula for compound interest is: [tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex] where: - [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest. - [tex]\( P \)[/tex] is the principal amount (initial investment). - [tex]\( r \)[/tex] is the annual interest rate (APR, in decimal form). - [tex]\( n \)[/tex] is the number of times the interest is compounded per year. - [tex]\( t \)[/tex] is the time the money is invested for in years.
Given: - [tex]\( P = 190 \)[/tex] (initial principal) - [tex]\( r = 0.02 \)[/tex] (2% annual interest rate, as a decimal) - [tex]\( n = 365 \)[/tex] (compounded daily)
Using these values in our formula:
[tex]\[ A = 190 \left(1 + \frac{0.02}{365}\right)^{365t} \][/tex]
Now, to simplify this we focus on the base of the exponent: [tex]\[ \left(1 + \frac{0.02}{365}\right)^{365} \][/tex]
Upon computing the above base, we get approximately [tex]\( 1.0202 \)[/tex]. So the function simplifies to: [tex]\[ A(t) = 190 \times 1.02020078103292^t \][/tex]
Rounded to four decimal places: [tex]\[ f(t) = 190 \times 1.0202^t \][/tex]
### Annual Percentage Yield (APY) APY is a way to express the annual growth rate of an investment with compounding interest. It can be calculated using the formula: [tex]\[ \text{APY} = \left(1 + \frac{r}{n}\right)^n - 1 \][/tex]
Substitute [tex]\( r = 0.02 \)[/tex] and [tex]\( n = 365 \)[/tex]: [tex]\[ \text{APY} = \left(1 + \frac{0.02}{365}\right)^{365} - 1 \][/tex]
This results in approximately [tex]\( 0.0202 \)[/tex] or [tex]\( 2.02\% \)[/tex].
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