To calculate the amount Zion would earn on a deposit of $10,000 at an interest rate of 5.12% compounded daily for one year, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = the annual interest rate (decimal).
- \( n \) = the number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested or borrowed.
Given:
- \( P = 10,000 \)
- \( r = 5.12% = 0.0512 \)
- \( n = 365 \) (compounded daily)
- \( t = 1 \)
Plugging in the values:
\[ A = 10,000 \left(1 + \frac{0.0512}{365}\right)^{365 \cdot 1} \]
Calculating:
- Calculate \( \frac{r}{n} \):
\[ \frac{0.0512}{365} \approx 0.0001408219 \]
- Calculate \( 1 + \frac{r}{n} \):
\[ 1 + 0.0001408219 \approx 1.0001408219 \]
- Calculate \( nt \):
\[ 365 \cdot 1 = 365 \]
- Now calculate:
\[ A = 10,000 \left(1.0001408219\right)^{365} \]
Using a calculator to find \( (1.0001408219)^{365} \):
\[ (1.0001408219)^{365} \approx 1.051712 \]
- Finally calculate \( A \):
\[ A \approx 10,000 \times 1.051712 \approx 10,517.12 \]
Now to find the earned interest:
\[ \text{Interest Earned} = A - P = 10,517.12 - 10,000 \approx 517.12 \]
Since the earnings closely match the options provided, we can round this to an appropriate amount.
The closest answer from the options given is $512.00.