Question

A deposit of $$\$ 7500$$ is made in an account that pays \(6 \%\) compounded quarterly. The amount \(A\) in the account after \(t\) years is \(A=7500(1.015)^{[4 t]}, \quad t \geq 0\) (a) Sketch the graph of \(A\). Is the graph continuous? Explain your reasoning. (b) What is the balance after 7 years?

Short answer

The graph of this formula is a smooth, upward sloping curve, indicating that it's continuous - this means that the account balance continuously increases over time without any breaks or gaps. After 7 years, the balance of the account will be the value calculated using the formula: \(A = 7500 * (1.015)^{28}\).

Step-by-step solution

Understanding the Formula

The formula \(A=7500(1.015)^{4t}\) is given by the problem. It shows the total amount \(A\) in the account after a period of \(t\) years, considering a starting amount (or 'principal') of $7500, an interest rate of 6% per year compounded quarterly (which means dividing the rate by 4 to get 1.5% or 0.015 per quarter), and a time \(t\) in years.

Sketching the Graph

Use the above-given formula to sketch the graph. Values of \(t\) will form the x-axis (as \(t\) represents time in years). Start it from 0 as stated in the problem. The values of \(A\) will form the y-axis (being the total account balance). The graph starts at $7500 when \(t=0\), then it increases exponentially as a result of the compounding interest effect, which is described by the \(1.015^{4t}\) part of the formula. This is because each quarter, the amount is multiplied by 1.015, thus accumulating more value over time. The graph plot of this formula will be a smooth, upward sloping curve. This indicates that the graph is in fact continuous, which means it has no break or gap.

Calculating the Balance After 7 Years

We use the provided formula with \(t=7\) to determine the account balance after 7 years. The result is computed as \(A = 7500 * (1.015)^{(4*7)}\)

Key concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a rate that is proportional to its current value. This means the bigger the quantity gets, the faster it grows. It's different from linear growth, where a constant amount is added over each time period. Exponential growth is characterized by the presence of a base raised to a power that includes the time variable, commonly seen in the formula for compound interest, such as in the given problem:

In our example, the starting deposit is \$7500\, and the growth is compounded quarterly. With a 6% annual interest rate divided into a 1.5% quarterly interest rate (0.015 as a decimal), the formula becomes \(A=7500(1.015)^{4t}\). The \(1.015^{4t}\) part of the equation is what signals the exponential growth of the investment over time.

Graphing Exponential Functions

Graphing exponential functions, like the one provided in our compound interest example, gives a visual representation of how the balance in an account changes over time. An exponential function graph typically starts out increasing slowly and then rapidly rises as time goes on.

To graph an exponential function such as \(A=7500(1.015)^{4t}\), you would plot time \(t\) along the x-axis and the account balance \(A\) along the y-axis. Starting with \(t=0\), which corresponds to the initial deposit, each quarter (every increase in \(t\) by 0.25), the amount would be multiplied by 1.015, thereby plotting points that rise swiftly as \(t\) increases. The result is a smooth curve that demonstrates the concept of continuous growth with no interruptions or steps, indicating the function's continuity.

Financial Mathematics

Financial mathematics is a field of applied mathematics concerned with financial markets and investments, and it typically uses formulas to calculate interest, analyze investments, and determine future values. Compound interest is a foundational concept in financial mathematics, as it is a practical application used to calculate the growth of investments over time.

In practice, to find the balance after a certain number of years using compound interest, we apply the formula provided by the financial institution. For example, after 7 years, we input 7 into our function for \(t\), resulting in \(A = 7500 * (1.015)^{(4*7)}\). The exponential function within this formula encapsulates the principle that money not only grows by accumulation of interest on the principal but also by earning interest on the interest already accumulated, which is known as compounding.