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Suppose that a tea company estimates that its monthly cost is [tex]$C(x)=500 x^2+100 x$[/tex] and its monthly revenue is [tex]$R(x)=-0.6 x^3+700 x^2-400 x+300$[/tex], where [tex][tex]$x$[/tex][/tex] is in thousands of boxes of tea sold. The profit is the difference between the revenue and the cost.

What is the profit function, [tex]$P(x)$[/tex]?

A. [tex]$P(x)=0.6 x^3-200 x^2+500 x-300$[/tex]

B. [tex][tex]$P(x)=-0.6 x^3+200 x^2-500 x+300$[/tex][/tex]

C. [tex]$P(x)=0.6 x^3+200 x^2-500 x+300$[/tex]

D. [tex]$P(x)=-0.6 x^3+1200 x^2-300 x+300$[/tex]



Answer :

GinnyAnswer
To find the profit function [tex]\( P(x) \)[/tex], we need to subtract the cost function [tex]\( C(x) \)[/tex] from the revenue function [tex]\( R(x) \)[/tex].

Given the functions:
[tex]\[ C(x) = 500x^2 + 100x \][/tex]
[tex]\[ R(x) = -0.6x^3 + 700x^2 - 400x + 300 \][/tex]

The profit function [tex]\( P(x) \)[/tex] is determined as follows:
[tex]\[ P(x) = R(x) - C(x) \][/tex]

Substitute [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into the equation:
[tex]\[ P(x) = (-0.6x^3 + 700x^2 - 400x + 300) - (500x^2 + 100x) \][/tex]

Now, distribute the negative sign across the terms in [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = -0.6x^3 + 700x^2 - 400x + 300 - 500x^2 - 100x \][/tex]

Combine like terms:
[tex]\[ P(x) = -0.6x^3 + (700x^2 - 500x^2) + (-400x - 100x) + 300 \][/tex]
[tex]\[ P(x) = -0.6x^3 + 200x^2 - 500x + 300 \][/tex]

Thus, the profit function is:
[tex]\[ P(x) = -0.6x^3 + 200x^2 - 500x + 300 \][/tex]

The correct choice is:
[tex]\[ \boxed{B. \, P(x) = -0.6x^3 + 200x^2 - 500x + 300} \][/tex]

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