Question

Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Savings account An initial deposit of \(\$ 1500\) is placed in a savings account with an APY of 3.1\%. How long will it take until the balance of the account is \(\$ 2500 ?\) Assume the interest rate remains constant and no additional deposits or withdrawals are made.

Short answer

Answer: It will take approximately 24.33 years for the balance to reach 2500 dollars.

Step-by-step solution

Determine the rate constant k

To find the rate constant k, we first need to convert the APY into a growth rate (r) in decimal form. To do this, we use the formula: r = APY/100. In this case, the APY is 3.1%, so r = 3.1/100 = 0.031.

Write the exponential growth function

The general form of an exponential growth function is given by: \(A(t) = A_0e^{kt}\), where \(A(t)\) is the amount in the account at time t, \(A_0\) is the initial deposit, k is the rate constant, and t is time (in years). In this case, we have A_0 = \(1500 and k = 0.031. Thus, our exponential growth function is: \)A(t) = 1500e^{0.031t}$.

Substitute the target balance amount and solve for t

We are given that the target balance is \(2500. To find the time it takes for the balance to reach this amount, we will substitute \)A(t) = 2500\( into our growth function and solve for t: \)2500 = 1500e^{0.031t}$.

Solve for t

We need to isolate t in the growth function equation. We can do this by dividing both sides by 1500: \(\frac{2500}{1500} = e^{0.031t}\). Next, take the natural logarithm (ln) of both sides: \(ln(\frac{2500}{1500}) = ln(e^{0.031t})\). Then, we can apply the logarithmic property that \(ln(a^b) = b*ln(a)\), which simplifies the equation to \(ln(\frac{2500}{1500}) = 0.031t\). Finally, solving for t, we get \(t = \frac{ln(\frac{2500}{1500})}{0.031}\).

Calculate the time (t) to reach the target balance

Using a calculator, we can find that \(\frac{ln(\frac{2500}{1500})}{0.031} \approx 24.33\). Therefore, it will take approximately 24.33 years for the balance in the savings account to reach 2500 dollars, assuming no additional deposits or withdrawals and a constant interest rate.

Key concepts

Rate Constant

The rate constant, often represented by "k" in exponential functions, holds a crucial role in determining how quickly quantities grow over time. When dealing with financial contexts, like savings accounts, it's the backbone of understanding exponential growth over the years.

To start, it's essential to convert percentages into their decimal forms, which can simplify calculations. In the context of a savings account with an APY of 3.1%, the rate (r) can be calculated as follows:

• Convert the percentage to a decimal: \( r = \frac{APY}{100} \)
• Thus, \( r = \frac{3.1}{100} = 0.031 \)

In the exponential growth formula \( A(t) = A_0e^{kt} \), "k" represents this very rate but in its purest form: the rate constant that drives the growth equation forward.

The primary equation to find our desired variable, time, is set once "k" is known. Understanding this enables us to unlock the logic behind how long it takes for amounts like savings to grow to specified future balances.

APY (Annual Percentage Yield)

The term APY stands for Annual Percentage Yield and indicates how much profit or interest one can earn on their money during the year. It's more than just a percentage—it summarizes the true return of an investment or savings over a complete year, considering compound interest.

Here's how APY plays out in our situation with a savings account:

• The initial amount, deposited at 3.1% APY, occasionally compounds, increasing the overall growth.
• Considering this APY is set against the backdrop of compounding, it highlights the factor by which the account grows annually.

Therefore, understanding APY is like having a sneak peek into how one's savings will evolve and why simply holding an account with an interest rate might not fully capture the potential gains over a financial year.

Savings Account

A savings account is a simple yet powerful financial tool that provides a safe place to keep money while earning interest over time. It embodies a financial buffer, encouraging savings, with the added benefit of accumulating interest as time progresses.

Let's break down how it works in relation to the given situation:

• The initial deposit, in this case, is $1500.
• It is characterized by a specific interest rate or APY, which, for this example, is 3.1%.
• No additional deposits or withdrawals impact the account, making it simpler to calculate its exponential growth.
• Because interest compounds, what starts as $1500, due to the power of compounded growth, can expand to a target like $2500 over time.

Armed with these attributes, a savings account becomes the perfect candidate to observe exponential growth in action. As the balance compounds, thanks to APY and the rate constant, it allows for visible and quantifiable financial growth—useful for both planning and achieving longer-term financial goals.