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The revenue, in dollars, of a company that produces jeans can be modeled by [tex]2x^2 + 17x - 175[/tex]. The cost, in dollars, of producing the jeans can be modeled by [tex]2x^2 - 3x - 125[/tex]. The number of pairs of jeans that have been sold is represented by [tex]x[/tex].

If the profit is the difference between the revenue and the cost, which expression can be used to find the profit and what is that profit when 75 pairs of jeans are sold?

A. [tex]20x - 50 ; \$500[/tex]
B. [tex]20x - 50 ; \$1,450[/tex]
C. [tex]20x + 50 ; \$1,550[/tex]
D. [tex]20x + 50 ; \$5,250[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to find the expression for profit and calculate the profit when 75 pairs of jeans are sold.

1. Define the revenue function [tex]\( R(x) \)[/tex]:
[tex]\[ R(x) = 2x^2 + 17x - 175 \][/tex]

2. Define the cost function [tex]\( C(x) \)[/tex]:
[tex]\[ C(x) = 2x^2 - 3x - 125 \][/tex]

3. Determine the profit function [tex]\( P(x) \)[/tex]:
Profit is the difference between revenue and cost:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions into the profit equation:
[tex]\[ P(x) = (2x^2 + 17x - 175) - (2x^2 - 3x - 125) \][/tex]

4. Simplify the profit function:
[tex]\[ P(x) = 2x^2 + 17x - 175 - 2x^2 + 3x + 125 \][/tex]
Combine like terms:
[tex]\[ P(x) = (2x^2 - 2x^2) + (17x + 3x) + (-175 + 125) \][/tex]
[tex]\[ P(x) = 20x - 50 \][/tex]
Therefore, the expression used to find profit is:
[tex]\[ P(x) = 20x - 50 \][/tex]

5. Calculate the profit when 75 pairs of jeans are sold:
Substitute [tex]\( x = 75 \)[/tex] into the profit function:
[tex]\[ P(75) = 20 \cdot 75 - 50 \][/tex]
[tex]\[ P(75) = 1500 - 50 \][/tex]
[tex]\[ P(75) = 1450 \][/tex]

So, the expression to find the profit is [tex]\( 20x - 50 \)[/tex], and when 75 pairs of jeans are sold, the profit is [tex]$1450. The correct choice is: \[ 20x - 50 ; \$[/tex] 1,450
\]