Question

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.

Accepted Answer

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

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Key Concepts

Exponential Growth Function

Rate Constant (k) in Continuous Growth

Solving for Time in Exponential Equations