15–20. 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.

Verified step by step guidance

1. Identify the exponential growth model for continuous compounding interest, which is given by the formula: \(A(t) = A_0 e^{k t}\), where \(A(t)\) is the amount at time \(t\), \(A_0\) is the initial amount, \(k\) is the rate constant, and \(t\) is time in years.
2. Use the given APY (Annual Percentage Yield) of 3.1% to find the rate constant \(k\). Since APY represents the effective annual growth, set \(A(1) = A_0 e^{k \cdot 1} = A_0 (1 + 0.031)\), and solve for \(k\) by isolating it in the equation \(e^{k} = 1.031\).
3. Write the exponential growth function using the initial deposit \(A_0 = 1500\) and the rate constant \(k\) found in the previous step: \(A(t) = 1500 e^{k t}\).
4. To find the time \(t\) when the balance reaches \$2500\(, set \)A(t) = 2500\( and solve for \)t\( in the equation \)2500 = 1500 e^{k t}$.
5. Isolate \(t\) by dividing both sides by 1500, then take the natural logarithm of both sides to get \(\ln\left(\frac{2500}{1500}\right) = k t\), and finally solve for \(t\) as \(t = \frac{\ln\left(\frac{2500}{1500}\right)}{k}\).

Key Concepts

Exponential Growth Function

An exponential growth function models quantities that increase at a rate proportional to their current value. It is generally expressed as A(t) = A_0 * e^(kt), where A_0 is the initial amount, k is the growth rate constant, and t is time. This function is essential for modeling continuous compound interest in savings accounts.

Rate Constant (k) in Continuous Growth

The rate constant k represents the continuous growth rate in the exponential model. It can be derived from the annual percentage yield (APY) using the formula k = ln(1 + APY). Knowing k allows us to write the exact exponential function that describes how the investment grows over time.

Solving for Time in Exponential Equations

To find the time required for an investment to reach a certain value, we solve the exponential equation for t. This involves isolating t by taking the natural logarithm of both sides, resulting in t = (1/k) * ln(A(t)/A_0). This step is crucial for answering questions about how long it takes for an investment to grow to a target amount.