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Profit is the difference between revenue and cost. The revenue, in dollars, of a company that manufactures cell phones can be modeled by the polynomial [tex]2x^2 + 55x + 10[/tex]. The cost, in dollars, of producing the cell phones can be modeled by [tex]2x^2 - 15x - 40[/tex]. The variable [tex]x[/tex] represents the number of cell phones sold.

What expression represents the profit, and what is the profit if 240 cell phones are sold?

A. [tex]40x - 30[/tex]; \$2,400
B. [tex]40x - 30[/tex]; \$9,570
C. [tex]70x + 50[/tex]; \$16,850
D. [tex]70x + 50[/tex]; \$28,800

Sagot :

GinnyAnswer
To solve this problem, let's break it down step by step.

1. Understanding the Problem:
- We are given the revenue function: [tex]\( R(x) = 2x^2 + 55x + 10 \)[/tex].
- We are also given the cost function: [tex]\( C(x) = 2x^2 - 15x - 40 \)[/tex].
- We need to find the profit function, [tex]\( P(x) \)[/tex], which is the difference between the revenue and the cost: [tex]\( P(x) = R(x) - C(x) \)[/tex].

2. Finding the Profit Function:
- The profit function is obtained by subtracting the cost function from the revenue function:
[tex]\[ P(x) = (2x^2 + 55x + 10) - (2x^2 - 15x - 40) \][/tex]
- We now distribute the negative sign through the cost function:
[tex]\[ P(x) = 2x^2 + 55x + 10 - 2x^2 + 15x + 40 \][/tex]
- Combine like terms:
[tex]\[ P(x) = (2x^2 - 2x^2) + (55x + 15x) + (10 + 40) \][/tex]
[tex]\[ P(x) = 0 + 70x + 50 \][/tex]
- Thus, the profit function simplifies to:
[tex]\[ P(x) = 70x + 50 \][/tex]

3. Calculating the Profit for 240 Cell Phones:
- To find the profit when [tex]\( x = 240 \)[/tex], we substitute [tex]\( x = 240 \)[/tex] into the profit function:
[tex]\[ P(240) = 70 \cdot 240 + 50 \][/tex]
- Simplify the expression:
[tex]\[ P(240) = 16800 + 50 \][/tex]
[tex]\[ P(240) = 16850 \][/tex]

4. Result:
- The profit function is [tex]\( 70x + 50 \)[/tex].
- The profit when 240 cell phones are sold is [tex]\( \$16850 \)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{70 x+50 ; \$ 16,850} \][/tex]
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