Select The Correct Answer From Each Drop-down Menu.The Function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4 T}$ Models The Balance In A Savings Account.The Savings Account Had An Initial Balance Of $\square$ And Compounds

The given function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4 t}$ represents the balance in a savings account. To understand this model, we need to break down the components and analyze each part.

Initial Balance

The initial balance of the savings account is the starting amount of money in the account. In this case, the initial balance is represented by the constant term in the function, which is $500$. This means that the account starts with a balance of $500.

Compounding Interest

Compounding interest is the process of earning interest on both the principal amount and any accrued interest. In this case, the interest rate is $0.015$, which is equivalent to $1.5\%$. The compounding frequency is $4$ times per year, which means that the interest is compounded quarterly.

Time

The time variable, $t$, represents the number of years that the money has been in the account. The function $f(x)$ models the balance in the account at any given time $t$.

Simplifying the Function

To simplify the function, we can evaluate the expression inside the parentheses:

$$\left(1+\frac{0.015}{4}\right)^{4 t} = \left(1+0.00375\right)^{4 t} = \left(1.00375\right)^{4 t}$$

This simplification makes it easier to understand the function and its behavior.

Understanding the Exponential Growth

The function $f(x)$ represents an exponential growth model, where the balance in the account grows at a rate proportional to the current balance. The exponential growth is represented by the term $\left(1.00375\right)^{4 t}$, which grows rapidly as $t$ increases.

Calculating the Balance

To calculate the balance in the account at any given time $t$, we can plug in the value of $t$ into the function $f(x)$. For example, if we want to calculate the balance after $5$ years, we can plug in $t=5$ into the function:

$$f(5) = 500\left(1.00375\right)^{4 \cdot 5}$$

Using a calculator or computer, we can evaluate this expression to find the balance after $5$ years.

Conclusion

In conclusion, the function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4 t}$ models the balance in a savings account with an initial balance of $500$, an interest rate of $1.5\%$, and compounding frequency of $4$ times per year. The function represents an exponential growth model, where the balance in the account grows rapidly as time increases. By understanding the components of the function and simplifying it, we can calculate the balance in the account at any given time.

Example Problems

Problem 1

What is the balance in the account after $10$ years?

Solution

To calculate the balance after $10$ years, we can plug in $t=10$ into the function $f(x)$:

$$f(10) = 500\left(1.00375\right)^{4 \cdot 10}$$

Using a calculator or computer, we can evaluate this expression to find the balance after $10$ years.

Problem 2

What is the interest rate in the account?

Solution

The interest rate in the account is $1.5\%$, which is equivalent to $0.015$.

Problem 3

What is the compounding frequency in the account?

Solution

The compounding frequency in the account is $4$ times per year.

Discussion Questions

1. What is the initial balance of the savings account?
2. What is the interest rate in the account?
3. What is the compounding frequency in the account?
4. How does the function $f(x)$ represent an exponential growth model?
5. How can we calculate the balance in the account at any given time?

Answer Key

1. The initial balance of the savings account is $500$.
2. The interest rate in the account is $1.5\%$, which is equivalent to $0.015$.
3. The compounding frequency in the account is $4$ times per year.
4. The function $f(x)$ represents an exponential growth model because the balance in the account grows at a rate proportional to the current balance.
5. We can calculate the balance in the account at any given time by plugging in the value of $t$ into the function $f(x)$.
   Q&A: Understanding the Savings Account Model =============================================

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

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

A: The initial balance of the savings account is $500$. This is the starting amount of money in the account.

Q: What is the interest rate in the account?

A: The interest rate in the account is $1.5\%$, which is equivalent to $0.015$.

Q: What is the compounding frequency in the account?

A: The compounding frequency in the account is $4$ times per year, which means that the interest is compounded quarterly.

Q: How does the function $f(x)$ represent an exponential growth model?

A: The function $f(x)$ represents an exponential growth model because the balance in the account grows at a rate proportional to the current balance. This is represented by the term $\left(1.00375\right)^{4 t}$, which grows rapidly as $t$ increases.

Q: How can we calculate the balance in the account at any given time?

A: We can calculate the balance in the account at any given time by plugging in the value of $t$ into the function $f(x)$. For example, if we want to calculate the balance after $5$ years, we can plug in $t=5$ into the function:

$$f(5) = 500\left(1.00375\right)^{4 \cdot 5}$$

Using a calculator or computer, we can evaluate this expression to find the balance after $5$ years.

Q: What is the significance of the exponent $4$ in the function $f(x)$?

A: The exponent $4$ in the function $f(x)$ represents the compounding frequency, which is $4$ times per year. This means that the interest is compounded quarterly, and the balance in the account grows rapidly as time increases.

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

A: Yes, we can use this model to calculate the balance in a savings account with a different interest rate. We simply need to replace the interest rate $0.015$ with the new interest rate in the function $f(x)$.

Q: How can we use this model to compare the performance of different savings accounts?

A: We can use this model to compare the performance of different savings accounts by plugging in the interest rates and compounding frequencies of each account into the function $f(x)$. This will give us the balance in each account at any given time, allowing us to compare their performance.

Q: What are some limitations of this model?

A: Some limitations of this model include:

• It assumes that the interest rate remains constant over time.
• It assumes that the compounding frequency remains constant over time.
• It does not take into account any fees or charges associated with the savings account.

Conclusion

In conclusion, the function $f(x)=500\left(1+\frac{0.015}{4}\right)^{4 t}$ models the balance in a savings account with an initial balance of $500$, an interest rate of $1.5\%$, and compounding frequency of $4$ times per year. We have answered some frequently asked questions about the model, and we have discussed its significance and limitations.

Example Problems

Problem 1

What is the balance in the account after $10$ years?

Solution

To calculate the balance after $10$ years, we can plug in $t=10$ into the function $f(x)$:

$$f(10) = 500\left(1.00375\right)^{4 \cdot 10}$$

Using a calculator or computer, we can evaluate this expression to find the balance after $10$ years.

Problem 2

What is the interest rate in the account?

Solution

The interest rate in the account is $1.5\%$, which is equivalent to $0.015$.

Problem 3

What is the compounding frequency in the account?

Solution

The compounding frequency in the account is $4$ times per year.

Discussion Questions

1. What is the initial balance of the savings account?
2. What is the interest rate in the account?
3. What is the compounding frequency in the account?
4. How does the function $f(x)$ represent an exponential growth model?
5. How can we calculate the balance in the account at any given time?

Answer Key

1. The initial balance of the savings account is $500$.
2. The interest rate in the account is $1.5\%$, which is equivalent to $0.015$.
3. The compounding frequency in the account is $4$ times per year.
4. The function $f(x)$ represents an exponential growth model because the balance in the account grows at a rate proportional to the current balance.
5. We can calculate the balance in the account at any given time by plugging in the value of $t$ into the function $f(x)$.