To analyze the expression \(500(1.00875)^{4t}\), we need to interpret it in terms of interest compounding:
- Base Amount: The $500 indicates the initial balance.
- Growth Factor: The term \(1.00875\) suggests that the account balance increases by \(0.875%\) for each compounding period.
- Exponent \(4t\): The exponent \(4t\) indicates that the interest is compounded quarterly since the factor \(1.00875\) is raised to \(4t\), where \(t\) is the number of years.
Next, let's derive the conditions under which this expression would represent the balance of a savings account:
Compounding Quarterly
Since the growth factor is \(1.00875\), we can find the annual interest rate that corresponds to this growth factor. For quarterly compounding, the effective growth factor for each quarter is given by the formula:
\[ 1 + \frac{r}{4} \]
where \(r\) is the annual interest rate. The quarterly interest rate here is \(0.875%\) (i.e. \(0.00875\)), we set up the equation:
\[ 1 + \frac{r}{4} = 1.00875 \]
Subtract \(1\):
\[ \frac{r}{4} = 0.00875 \]
Multiply both sides by \(4\):
\[ r = 0.00875 \times 4 = 0.035 = 3.5% \]
Conclusion
Thus, the expression \(500(1.00875)^{4t}\) represents the balance of a savings account after \(t\) years if the initial balance is $500 and it is invested at a 3.5% annual interest rate compounded quarterly.
Comparing this conclusion to the given options:
- A is incorrect because it states annual compounding.
- B is incorrect because it states \(0.875%\) annual rate instead of \(3.5%\).
- C is incorrect because the interest rate stated is not \(3.5%\) and clarifies compounding quarterly.
- D is correct, stating \(3.5%\) and compounding quarterly.
Thus, the correct answer is
\[ \boxed{D} \]