To find the rate at which the balance is changing at different times $t$, we need to determine the derivative of the balance function $A(t)$ with respect to time $t$. This derivative will give us the rate of change of the balance at any given time $t$.

Step 1: Given equation

The balance function is: $A(t) = 8000 \cdot e^{0.06t}$

Step 2: Find the derivative of $A(t)$

We take the derivative of $A(t)$ with respect to $t$: $\frac{dA}{dt} = 8000 \cdot \frac{d}{dt} \left( e^{0.06t} \right)$ Using the chain rule, the derivative of $e^{0.06t}$ is $0.06 \cdot e^{0.06t}$, so: $\frac{dA}{dt} = 8000 \cdot 0.06 \cdot e^{0.06t}$ $\frac{dA}{dt} = 480 \cdot e^{0.06t}$

Step 3: Evaluate the rate of change at different times $t$

1. For $t = 1$: $\frac{dA}{dt} = 480 \cdot e^{0.06 \cdot 1} = 480 \cdot e^{0.06}$ Using $e^{0.06} \approx 1.06184$, we get: $\frac{dA}{dt} \approx 480 \cdot 1.06184 \approx 509.68$ So, the rate of change at $t = 1$ year is approximately 509.68 dollars/year.

2. For $t = 10$: $\frac{dA}{dt} = 480 \cdot e^{0.06 \cdot 10} = 480 \cdot e^{0.6}$ Using $e^{0.6} \approx 1.82212$, we get: $\frac{dA}{dt} \approx 480 \cdot 1.82212 \approx 874.62$ So, the rate of change at $t = 10$ years is approximately 874.62 dollars/year.

3. For $t = 50$: $\frac{dA}{dt} = 480 \cdot e^{0.06 \cdot 50} = 480 \cdot e^{3}$ Using $e^{3} \approx 20.0855$, we get: $\frac{dA}{dt} \approx 480 \cdot 20.0855 \approx 9641.04$ So, the rate of change at $t = 50$ years is approximately 9641.04 dollars/year.

Summary of results:

• Rate of change at $t = 1$ year: 509.68 dollars/year
• Rate of change at $t = 10$ years: 874.62 dollars/year
• Rate of change at $t = 50$ years: 9641.04 dollars/year

Would you like further details on this solution or have any questions?

5 Related Questions:

1. What would the balance be at $t = 1$, $t = 10$, and $t = 50$ years?
2. How does the rate of change compare with the initial deposit?
3. How would the rate of change differ if the interest rate were 5% instead of 6%?
4. Can we approximate the time it takes for the balance to double?
5. What would the derivative look like if the interest compounded quarterly?

Tip: When dealing with exponential growth, small changes in time can lead to increasingly larger rates of change, especially as time increases.