From the question we are given
[tex]\text{Profit }=Total\text{ Revenue }-total\text{ cost}[/tex]
Annual total cost in $ is given by
[tex]C(x)=3600+100x+2x^2_{}[/tex]
And an Annual total revenue in $ is given as
[tex]R(x)=500x-2x^2[/tex]
Where x is the number of pairs of boots sold
We are to find the profit function
Using the detains given we have
Profit function P(x) is
[tex]P(x)=R(x)-C(x)[/tex]
Therefore,
[tex]\begin{gathered} P(x)=500x-2x^2-\lbrack3600+100x+2x^2\rbrack \\ P(x)=500x-2x^2-3600-100x-2x^2 \\ P(x)=-4x^2+400x-3600 \end{gathered}[/tex]
Therefore, the profit function is
[tex]P(x)=-4x^2+400x-3600[/tex]
Next we are to find determine the number of pair of boots that will maximize annual profit
Plotting the graph of the Profit function we have
Hence, at maximum we have the points (50, 6400)
Therefore, x = 50
This implies that the number of boots that will maximize the annual profit is 500
