Question

"Using the function f(x) = 500(1 + 0.015)^4t, which models the balance in a savings account that had an initial balance of $500 and compounds quarterly at an interest rate of 1.50%, calculate the account balance after a specified time period.

Answer 1

The balance in a savings account with an initial deposit of $500 that compounds quarterly at an interest rate of 1.50% can be calculated using the compound interest formula. By plugging the number of years 't' into the formula, you can determine the account balance after that time.

Step-by-step explanation:

Calculating Compound Interest

To find the balance in a savings account after a certain period of time with a given interest rate that compounds quarterly, we use the compound interest formula. The compound interest formula is:

Principal(1 + interest rate)number of times interest is compounded per year×time

In this case, the initial balance is $500, the quarterly interest rate is 1.50% (0.015), and the number of times the interest is compounded per year is 4, since it is compounded quarterly. If we want to calculate the balance after 't' years, we substitute these values into the formula as follows:

f(x) = 500(1 + 0.015)4t

We simply have to know the number of years 't' to find the final balance.

For example, if we wanted to calculate the balance after 5 years, we would plug in 't' as 5:

f(x) = 500(1 + 0.015)4×5
= 500(1.015)20
= 500(1.346855)

Then, the balance would be:

f(x) = 500 × 1.346855 ≈ $673.43 after 5 years.

This calculation demonstrates the effect of compound interest over time, showing that the initial deposit grows as interest is added periodically at the set rate.

Answer 2

To calculate the account balance after a specific time using the formula f(x) = 500(1 + 0.015)^4t, replace t with the number of years and compute the result. This applies to an account with quarterly compound interest, not simple interest, which has a different formula.

Step-by-step explanation:

Calculating Compound Interest

To calculate the balance in the savings account after a certain period, you will use the compound interest formula given by f(x) = 500(1 + 0.015)^4t, where 1.50% is the quarterly interest rate and t is the number of years. To find out the account balance after a specified time period, you need to substitute the value for t (in years) into the formula.

Example Calculation

If you want to find the balance after 5 years, you will:

Set t to 5.

Calculate the exponent part: (1 + 0.015)^(4*5) = (1 + 0.015)^20.

Use a calculator to determine the result of (1 + 0.015)^20.

Multiply the initial principal (which is $500) by the result of the previous step to find the final balance.

Remember, this formula applies to accounts with quarterly compounding interest. If the question instead involves simple interest, you would use a different formula, which is given by Interest = Principal × rate × time. However, this is not applicable in the current scenario.