To find the average weekly sales over the first 5 weeks, we calculate the average value of the function $S(t)$ over the interval $[0, 5]$.

The formula for the average value of a function $f(t)$ over $[a, b]$ is:

$\text{Average value} = \frac{1}{b-a} \int_a^b f(t) \, dt$

For the given equation $S(t) = 5e^t$, $a = 0$, and $b = 5$, the calculation becomes:

$\text{Average sales} = \frac{1}{5-0} \int_0^5 5e^t \, dt$

Steps to Solve:

1. Compute the integral of $5e^t$: $\int 5e^t \, dt = 5e^t + C$

2. Evaluate the definite integral from $t = 0$ to $t = 5$: $\int_0^5 5e^t \, dt = \left[ 5e^t \right]_0^5 = 5e^5 - 5e^0$

3. Simplify: $5e^0 = 5 \quad \text{and} \quad 5e^5 = 5 \times e^5$ Substitute $e^5 \approx 148.413$: $5e^5 \approx 5 \times 148.413 = 742.065$ $\int_0^5 5e^t \, dt \approx 742.065 - 5 = 737.065$

4. Divide by 5 to find the average: $\text{Average sales} = \frac{737.065}{5} \approx 147.41 \, (\text{hundreds of dollars})$

5. Convert to dollars: $\text{Average sales} = 147.41 \times 100 = 14,741.00$

Final Answer:

The average weekly sales for the first 5 weeks is approximately $14,741.00.

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Would you like further explanation on these steps or the concept of average value?

Related Questions:

1. What is the instantaneous rate of sales at $t = 5$?
2. How does the average sales change if calculated for 10 weeks instead of 5?
3. Can the equation be approximated with simpler exponential growth patterns?
4. What is the total sales revenue accumulated in the first 5 weeks?
5. How would the average change if the function had a coefficient other than 5?

Tip: When calculating definite integrals involving $e^t$, remember that the exponential function is one of the easiest to differentiate and integrate.