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]
