Solution to Example Questions

Oct 13, 2025 by leolavanderia.com.br 145 views

The given function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4t}$ represents the balance in a savings account over time. To gain a deeper understanding of this model, we need to analyze its components and extract relevant information.

Initial Balance

The initial balance of the savings account can be determined by evaluating the function at $t=0$ . Substituting $t=0$ into the function, we get:

$f(0) = 500\left(1+\frac{0.015}{4}\right)^{4(0)}$

Simplifying the expression, we get:

$f(0) = 500\left(1+\frac{0.015}{4}\right)^0$

Since any number raised to the power of 0 is equal to 1, we have:

$f(0) = 500(1)^0$

$f(0) = 500$

Therefore, the initial balance of the savings account is $500.

Compounding Frequency

The compounding frequency of the interest can be determined by analyzing the exponent of the function. The exponent is $4t$ , which indicates that the interest is compounded quarterly. This is because the interest is compounded 4 times in a year, resulting in a total of 4 compounding periods.

Interest Rate

The interest rate can be determined by analyzing the coefficient of the exponent. The coefficient is $\frac{0.015}{4}$ , which represents the interest rate per compounding period. To find the annual interest rate, we need to multiply this value by 4:

$Annual\ Interest\ Rate = 4 \times \frac{0.015}{4}$

$Annual\ Interest\ Rate = 0.015$

Therefore, the annual interest rate is 1.5%.

Conclusion

In conclusion, the savings account had an initial balance of $500 and compounds quarterly at an interest rate of 1.5%. This information can be used to make informed decisions about the savings account and to plan for future financial goals.

Example Questions

What is the balance in the savings account after 5 years, assuming the interest rate remains constant?
How much interest is earned in the first year, assuming the interest rate remains constant?
What is the balance in the savings account after 10 years, assuming the interest rate remains constant?

Solution to Example Questions

To find the balance in the savings account after 5 years, we need to substitute $t=5$ into the function:

$f(5) = 500\left(1+\frac{0.015}{4}\right)^{4(5)}$

Simplifying the expression, we get:

$f(5) = 500\left(1+\frac{0.015}{4}\right)^{20}$

Using a calculator to evaluate the expression, we get:

$f(5) = 500(1.00375)^{20}$

$f(5) = 500(1.0773)$

$f(5) = 538.65$

Therefore, the balance in the savings account after 5 years is $538.65.

To find the interest earned in the first year, we need to subtract the initial balance from the balance after 1 year:

$f(1) = 500\left(1+\frac{0.015}{4}\right)^{4(1)}$

Simplifying the expression, we get:

$f(1) = 500\left(1+\frac{0.015}{4}\right)^4$

Using a calculator to evaluate the expression, we get:

$f(1) = 500(1.00375)^4$

$f(1) = 500(1.0153)$

$f(1) = 507.65$

Therefore, the interest earned in the first year is $7.65.

To find the balance in the savings account after 10 years, we need to substitute $t=10$ into the function:

$f(10) = 500\left(1+\frac{0.015}{4}\right)^{4(10)}$

Simplifying the expression, we get:

$f(10) = 500\left(1+\frac{0.015}{4}\right)^{40}$

Using a calculator to evaluate the expression, we get:

$f(10) = 500(1.00375)^{40}$

$f(10) = 500(1.1619)$

$f(10) = 580.95$

In this article, we will answer some frequently asked questions about the savings account model represented by the function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4t}$ .

Q: What is the formula for calculating the balance in the savings account?

A: The formula for calculating the balance in the savings account is given by the function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4t}$ .

Q: How do I calculate the balance in the savings account after a certain number of years?

A: To calculate the balance in the savings account after a certain number of years, you need to substitute the number of years into the function. For example, to find the balance after 5 years, you would substitute $t=5$ into the function.

Q: What is the interest rate used in the savings account model?

A: The interest rate used in the savings account model is 1.5% per annum, compounded quarterly.

Q: How often is the interest compounded in the savings account model?

A: The interest is compounded quarterly in the savings account model.

Q: What is the initial balance of the savings account?

A: The initial balance of the savings account is $500.

Q: How much interest is earned in the first year?

A: To find the interest earned in the first year, you need to subtract the initial balance from the balance after 1 year. Using the function, we get:

$f(1) = 500\left(1+\frac{0.015}{4}\right)^4$

$f(1) = 507.65$

Therefore, the interest earned in the first year is $7.65.

Q: How much will the balance in the savings account be after 10 years?

A: To find the balance in the savings account after 10 years, you need to substitute $t=10$ into the function:

$f(10) = 500\left(1+\frac{0.015}{4}\right)^{40}$

Using a calculator to evaluate the expression, we get:

$f(10) = 500(1.00375)^{40}$

$f(10) = 500(1.1619)$

$f(10) = 580.95$

Therefore, the balance in the savings account after 10 years is $580.95.

Q: Can I use this model to calculate the balance in a savings account with a different interest rate?

A: Yes, you can use this model to calculate the balance in a savings account with a different interest rate. Simply substitute the new interest rate into the function and calculate the balance.

Q: How accurate is this model?

A: This model is an approximation of the balance in a savings account and assumes that the interest rate remains constant over time. In reality, interest rates may fluctuate, and other factors such as fees and taxes may affect the balance. Therefore, this model should be used as a rough estimate only.

Q: Can I use this model to calculate the balance in a savings account with a different compounding frequency?

A: Yes, you can use this model to calculate the balance in a savings account with a different compounding frequency. Simply substitute the new compounding frequency into the function and calculate the balance.

Q: How do I calculate the interest earned in a savings account with a different compounding frequency?

A: To calculate the interest earned in a savings account with a different compounding frequency, you need to use the formula for compound interest:

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

Where:

A is the balance in the savings account
P is the principal (initial balance)
r is the interest rate
n is the number of compounding periods per year
t is the number of years

Using this formula, you can calculate the interest earned in a savings account with a different compounding frequency.