You deposit $750 in a savings account that earns 5% annual interest compounded quarterly. a. Write a function that represents the balance y (in dollars) after t years. b. What is t... You deposit $750 in a savings account that earns 5% annual interest compounded quarterly. a. Write a function that represents the balance y (in dollars) after t years. b. What is the balance of the account after 4 years? Read more

Steps to Solve

1. Identify the Compounding Interest Formula

To find the balance in a savings account that compounds interest, we use the formula:

$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$

where:

• ( A ) = the amount of money accumulated after n years, including interest.
• ( P ) = the principal amount (the initial amount of money).
• ( r ) = annual interest rate (decimal).
• ( n ) = number of times that interest is compounded per year.
• ( t ) = the number of years the money is invested or borrowed.

2. Substitute Known Values into the Formula

Given:

• ( P = 750 )
• ( r = 0.05 )
• ( n = 4 ) (quarterly compounding)
• ( t = 4 )

Substituting these values in, we have:

$$ A = 750 \left(1 + \frac{0.05}{4}\right)^{4 \cdot t} $$

3. Simplify the Equation

Calculate the term inside the parentheses:

$$ A = 750 \left(1 + \frac{0.05}{4}\right)^{4t} = 750 \left(1 + 0.0125\right)^{4t} = 750 (1.0125)^{4t} $$

Now, substituting ( t ) with 4:

$$ A = 750 (1.0125)^{16} $$

4. Calculate the Final Balance

Now, we calculate ( (1.0125)^{16} ):

Using a calculator,

$$ (1.0125)^{16} \approx 1.221386025 $$

Therefore, we compute:

$$ A \approx 750 \times 1.221386025 \approx 915.99 $$