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Profit is the difference between revenue and cost. The revenue, in dollars, of a company that makes skateboards can be modeled by the polynomial [tex]2x^3 + 30x - 130[/tex]. The cost, in dollars, of producing the skateboards can be modeled by [tex]2x^3 - 3x - 520[/tex]. The variable [tex]x[/tex] represents the number of skateboards sold.

What expression represents the profit?

A. [tex]27x - 650[/tex]

B. [tex]27x + 390[/tex]

C. [tex]33x - 650[/tex]

D. [tex]33x + 390[/tex]



Answer :

GinnyAnswer
To determine the profit expression, we need to find the difference between the revenue and the cost. Let's start by writing down the expressions given for the revenue and the cost.

1. Revenue (R):
[tex]\[ R(x) = 2x^3 + 30x - 130 \][/tex]

2. Cost (C):
[tex]\[ C(x) = 2x^3 - 3x - 520 \][/tex]

The profit (P) is defined as the difference between the revenue and the cost:
[tex]\[ P(x) = R(x) - C(x) \][/tex]

Substitute the expressions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = (2x^3 + 30x - 130) - (2x^3 - 3x - 520) \][/tex]

Let's simplify this step-by-step:

1. Distribute the negative sign through the cost expression:
[tex]\[ P(x) = 2x^3 + 30x - 130 - 2x^3 + 3x + 520 \][/tex]

2. Combine like terms:
- Combine [tex]\( 2x^3 \)[/tex] and [tex]\( -2x^3 \)[/tex]:
[tex]\[ 2x^3 - 2x^3 = 0 \][/tex]

- Combine [tex]\( 30x \)[/tex] and [tex]\( 3x \)[/tex]:
[tex]\[ 30x + 3x = 33x \][/tex]

- Combine [tex]\( -130 \)[/tex] and [tex]\( 520 \)[/tex]:
[tex]\[ -130 + 520 = 390 \][/tex]

So, the simplified expression for profit is:
[tex]\[ P(x) = 33x + 390 \][/tex]

Hence, the correct choice that represents the profit is:
[tex]\[ \boxed{33x + 390} \][/tex]