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The total profit made by an engineering firm is given by the function p=x^2 - 24x +5000 where x is the number of clients the firm has and p is the profit.


1. Find the maximum profit made by the company.

2. How many clients are necessary to reach the profit outlined in part one?

Sagot :

Boffeemadrididn

Answer:

[tex]5144[/tex]

[tex]12[/tex]

Step-by-step explanation:

The function [tex]p=x^2-24x+5000[/tex] is incorrect as its roots are imaginary [tex]b^2-4ac=576-4\times 5000=-19424<0[/tex].

So, the correct function is

[tex]p=-x^2+24x+5000[/tex]

Differentiating with respect to [tex]x[/tex] we get

[tex]p'=-2x+24[/tex]

Equating with zero

[tex]0=-2x+24\\\Rightarrow x=\dfrac{24}{2}\\\Rightarrow x=12[/tex]

Double derivative of the function

[tex]p''=-2<0[/tex]

So, the function is maximum at [tex]x=12[/tex]

Maximum profit is

[tex]p=-x^2+24x+5000=-12^2+24\times 12+5000\\\Rightarrow p=5144[/tex]

The maximum profit made by the company is [tex]5144[/tex]

The number of clients required to make the maximum profit is [tex]12[/tex].

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