Question

a company determines that its weekly online sales, ( s(t) ), in hundreds of dollars, ( t ) weeks after online sales began can be estimated by ( s(t)=62 e^{t} ). find the average weekly sales for the first 7 weeks after online sales began.

Explanation:

Step1: Recall the formula for average value

The formula for the average value of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{1}{b - a}\int_{a}^{b}f(x)dx\). Here, \(a = 0\), \(b=7\), and \(S(t)=62e^{t}\). So the average weekly sales \(\bar{S}=\frac{1}{7-0}\int_{0}^{7}62e^{t}dt\).

Step2: Integrate the function

We know that \(\int e^{t}dt=e^{t}+C\). Then \(\frac{62}{7}\int_{0}^{7}e^{t}dt=\frac{62}{7}[e^{t}]_{0}^{7}\).

Step3: Evaluate the definite - integral

Using the fundamental theorem of calculus \(F(b)-F(a)\) (where \(F(t)\) is an antiderivative of \(f(t)\)), we have \(\frac{62}{7}(e^{7}-e^{0})\). Since \(e^{0} = 1\), and \(e^{7}\approx1096.633\), then \(\frac{62}{7}(1096.633 - 1)=\frac{62\times1095.633}{7}\).

Step4: Calculate the final value

\(\frac{62\times1095.633}{7}=\frac{67929.246}{7}\approx9704.18\). Since \(S(t)\) is in hundreds of dollars, the average weekly sales is \(9704.18\times100 = 970418\) dollars.

Answer:

\(970418\) dollars