Let's break this question into two parts: constructing the function and calculating the annual percentage yield (APY).
### Step 1: Constructing the Function Showing the Value of the Account
We know the investment is $8100, the annual interest rate (APR) is 3.7%, and the interest is compounded daily.
1. Initial investment [tex]\( P \)[/tex]:
[tex]\[ P = 8100 \][/tex]
2. Annual interest rate (APR):
[tex]\[ r = 3.7\% = 0.037 \][/tex]
3. Number of times the interest is compounded per year:
[tex]\[ n = 365 \][/tex]
The formula for the value of an investment compounded [tex]\( n \)[/tex] times per year is:
[tex]\[
A(t) = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Substituting the known values:
[tex]\[
A(t) = 8100 \left(1 + \frac{0.037}{365}\right)^{365t}
\][/tex]
Next, we need to round the coefficients to four decimal places:
Let’s denote:
[tex]\[
1 + \frac{0.037}{365} = 1 + 0.0001014 \approx 1.0001
\][/tex]
Therefore, the function [tex]\( f \)[/tex] showing the value of the account after [tex]\( t \)[/tex] years is:
[tex]\[
f(t) = 8100 \times 1.0001^{365t}
\][/tex]
### Step 2: Determining the APY (Annual Percentage Yield)
APY is the effective annual rate. It can be calculated using the formula:
[tex]\[
APY = \left(1 + \frac{r}{n}\right)^n - 1
\][/tex]
Substituting the known values:
[tex]\[
APY = \left(1 + \frac{0.037}{365}\right)^{365} - 1
\][/tex]
Upon calculation and rounding to the nearest hundredth of a percent:
[tex]\[
APY \approx 3.77\%
\][/tex]
### Final Answer
The function showing the value of the account after [tex]\( t \)[/tex] years, with coefficients rounded to four decimal places, is:
[tex]\[
f(t) = 8100 \times 1.0001^{365t}
\][/tex]
The percentage of growth per year (APY) to the nearest hundredth of a percent is:
[tex]\[
3.77\%
\][/tex]
