To determine Wacky Widget Company's profit for the month when they produce and sell 100 widgets, we need to follow the steps using the given information. Let's use the revenue function, cost function, and profit function provided.
1. Revenue Function:
The revenue function, denoted as [tex]\(r(w)\)[/tex], is given by:
[tex]\[
r(w) = 90w
\][/tex]
Here, [tex]\(w\)[/tex] represents the number of widgets produced and sold. For 100 widgets, we plug [tex]\(w = 100\)[/tex] into the revenue function:
[tex]\[
r(100) = 90 \times 100 = 9,000
\][/tex]
2. Cost Function:
The cost function, denoted as [tex]\(c(w)\)[/tex], is given by:
[tex]\[
c(w) = 57w + 5,000
\][/tex]
Again, for 100 widgets, we use [tex]\(w = 100\)[/tex]:
[tex]\[
c(100) = 57 \times 100 + 5,000
\][/tex]
Calculating this:
[tex]\[
c(100) = 5,700 + 5,000 = 10,700
\][/tex]
3. Profit Function:
The profit function, denoted as [tex]\(p(w)\)[/tex], is determined by subtracting the cost from the revenue:
[tex]\[
p(w) = r(w) - c(w)
\][/tex]
For 100 widgets, we substitute the previously calculated values of [tex]\(r(100)\)[/tex] and [tex]\(c(100)\)[/tex]:
[tex]\[
p(100) = r(100) - c(100) = 9,000 - 10,700
\][/tex]
Simplifying this gives:
[tex]\[
p(100) = -1,700
\][/tex]
Thus, Wacky Widget Company's profit for the month when producing and selling 100 widgets is [tex]\(\$ -1,700\)[/tex]. This indicates a loss of \$1,700.
