Answer :
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].
### Final Answer
Putting this all together:
Function:
[tex]\[ f(t) = 190 \times 1.0202^t \][/tex]
Percentage Growth per Year (APY):
[tex]\[ 2.02\% \][/tex]
This gives us the final formula and the percentage growth per year for the given investment.
### 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].
### Final Answer
Putting this all together:
Function:
[tex]\[ f(t) = 190 \times 1.0202^t \][/tex]
Percentage Growth per Year (APY):
[tex]\[ 2.02\% \][/tex]
This gives us the final formula and the percentage growth per year for the given investment.
