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Suppose that a guitar company estimates that its monthly cost is [tex]$C(x)=400x^2+600x[tex]$[/tex] and its monthly revenue is [tex]$[/tex]R(x)=-0.4x^3+600x^2-200x+500[tex]$[/tex], where [tex]$[/tex]x$[/tex] is in thousands of guitars sold. The profit is the difference between the revenue and the cost.

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

A. [tex]$P(x)=-0.4x^3+1000x^2+400x+500$[/tex]
B. [tex]$P(x)=0.4x^3+200x^2-800x+500$[/tex]
C. [tex]$P(x)=-0.4x^3+200x^2-800x+500$[/tex]
D. [tex]$P(x)=0.4x^3-200x^2+800x-500$[/tex]

Sagot :

GinnyAnswer
To find the profit function \( P(x) \), we need to calculate the difference between the revenue function \( R(x) \) and the cost function \( C(x) \). The given functions are defined as follows:

[tex]\[ C(x) = 400x^2 + 600x \][/tex]
[tex]\[ R(x) = -0.4x^3 + 600x^2 - 200x + 500 \][/tex]

The profit function \( P(x) \) is:

[tex]\[ P(x) = R(x) - C(x) \][/tex]

Substituting the given functions, we get:

[tex]\[ P(x) = (-0.4x^3 + 600x^2 - 200x + 500) - (400x^2 + 600x) \][/tex]

Distribute the negative sign to the terms in the cost function:

[tex]\[ P(x) = -0.4x^3 + 600x^2 - 200x + 500 - 400x^2 - 600x \][/tex]

Combine like terms:

- The \( x^3 \) term: \(-0.4x^3\)
- The \( x^2 \) terms: \(600x^2 - 400x^2 = 200x^2\)
- The \( x \) terms: \(-200x - 600x = -800x\)
- The constant term: \(500\)

So, the profit function is:

[tex]\[ P(x) = -0.4x^3 + 200x^2 - 800x + 500 \][/tex]

Now, let's verify which option matches our derived profit function:

A. \( P(x) = -0.4x^3 + 1000x^2 + 400x + 500 \)
B. \( P(x) = 0.4x^3 + 200x^2 - 800x + 500 \)
C. \( P(x) = -0.4x^3 + 200x^2 - 800x + 500 \)
D. \( P(x) = 0.4x^3 - 200x^2 + 800x - 500 \)

The correct answer is:

C. [tex]\( P(x) = -0.4x^3 + 200x^2 - 800x + 500 \)[/tex]
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