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Example 3

The Wacky Widget Company is planning to produce widgets. The company has rented space for its manufacturing operation at \[tex]$5,000 per month. Each widget requires \$[/tex]35 worth of materials and \$22 worth of labor.

Recall the revenue function and the cost function for Wacky Widgets:
[tex]\[
\begin{array}{l}
r(w) = 90w \\
c(w) = 57w + 5,000
\end{array}
\][/tex]

Since [tex]\( p(w) = r(w) - c(w) \)[/tex], find the function that Wacky Widgets can use to calculate its monthly profit.

Click on the correct answer.
[tex]\[
\begin{array}{l}
p(w) = 90w - (57w + 5,000) \\
p(w) = (57w + 5,000) - 90w \\
p(w) = (57w + 5,000) + 90w \\
p(w) = 90w + (57w + 5,000)
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
To determine the monthly profit function [tex]\( p(w) \)[/tex] for Wacky Widgets, we need to subtract the cost function [tex]\( c(w) \)[/tex] from the revenue function [tex]\( r(w) \)[/tex].

Given:
- The revenue function [tex]\( r(w) = 90w \)[/tex]
- The cost function [tex]\( c(w) = 57w + 5,000 \)[/tex]

We know that profit [tex]\( p(w) \)[/tex] is defined as the revenue minus the cost:
[tex]\[ p(w) = r(w) - c(w) \][/tex]

Substitute the given functions into the profit equation:
[tex]\[ p(w) = 90w - (57w + 5,000) \][/tex]

Now, simplify the expression inside the parentheses:
[tex]\[ p(w) = 90w - 57w - 5,000 \][/tex]

Combine like terms:
[tex]\[ p(w) = (90 - 57)w - 5,000 \][/tex]

Perform the subtraction:
[tex]\[ p(w) = 33w - 5,000 \][/tex]

Therefore, the function that Wacky Widgets can use to calculate its monthly profit is:
[tex]\[ p(w) = 90w - (57w + 5,000) \][/tex]

The correct answer is:
[tex]\[ p(w) = 90w - (57w + 5,000) \][/tex]
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