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A magazine company estimates that its monthly cost is [tex]C(x)=700 x^2+400 x[/tex] and its monthly revenue is [tex]R(x)=-0.4 x^3+900 x^2-200 x+600[/tex], where [tex]x[/tex] is in thousands of magazines 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)=-0.4 x^3+200 x^2-600 x+600[/tex]
B. [tex]P(x)=0.4 x^3+200 x^2-600 x+600[/tex]
C. [tex]P(x)=0.4 x^3-200 x^2+600 x-600[/tex]
D. [tex]P(x)=-0.4 x^3+1600 x^2+200 x+600[/tex]



Answer :

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

The given functions are:
- Cost function: [tex]\( C(x) = 700 x^2 + 400 x \)[/tex]
- Revenue function: [tex]\( R(x) = -0.4 x^3 + 900 x^2 - 200 x + 600 \)[/tex]

The profit function [tex]\( P(x) \)[/tex] is given by:
[tex]\[ P(x) = R(x) - C(x) \][/tex]

Substitute the given functions into the equation for profit:
[tex]\[ P(x) = \left(-0.4 x^3 + 900 x^2 - 200 x + 600\right) - \left(700 x^2 + 400 x\right) \][/tex]

Now, distribute the negative sign in the cost function:
[tex]\[ P(x) = -0.4 x^3 + 900 x^2 - 200 x + 600 - 700 x^2 - 400 x \][/tex]

Combine like terms:
[tex]\[ P(x) = -0.4 x^3 + (900 x^2 - 700 x^2) + (-200 x - 400 x) + 600 \][/tex]
[tex]\[ P(x) = -0.4 x^3 + 200 x^2 - 600 x + 600 \][/tex]

Therefore, the profit function is:
[tex]\[ P(x) = -0.4 x^3 + 200 x^2 - 600 x + 600 \][/tex]

This matches option A.

Thus, the correct answer is:
A. [tex]\( P(x) = -0.4 x^3 + 200 x^2 - 600 x + 600 \)[/tex]

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