Answer :
To determine the correct expression for the profit and the profit value when 75 pairs of jeans are sold, follow these steps:
### Step-by-Step Solution
1. Identify the expressions for revenue and cost:
The revenue function is given by:
[tex]\[ \text{Revenue} = 2x^2 + 17x - 175 \][/tex]
The cost function is given by:
[tex]\[ \text{Cost} = 2x^2 - 3x - 125 \][/tex]
2. Formulate the profit function:
Profit is calculated as the difference between revenue and cost.
[tex]\[ \text{Profit} = \text{Revenue} - \text{Cost} \][/tex]
Substitute the given expressions:
[tex]\[ \text{Profit} = (2x^2 + 17x - 175) - (2x^2 - 3x - 125) \][/tex]
3. Simplify the profit expression:
Combine like terms:
[tex]\[ \text{Profit} = 2x^2 + 17x - 175 - 2x^2 + 3x + 125 \][/tex]
Simplify further by combining [tex]\(2x^2\)[/tex] terms which cancel each other out:
[tex]\[ \text{Profit} = 17x + 3x - 175 + 125 \][/tex]
[tex]\[ \text{Profit} = 20x - 50 \][/tex]
Therefore, the expression for profit is:
[tex]\[ \text{Profit} = 20x - 50 \][/tex]
4. Calculate the profit when [tex]\( x = 75 \)[/tex]:
Substitute [tex]\( x = 75 \)[/tex] into the profit expression:
[tex]\[ \text{Profit} = 20(75) - 50 \][/tex]
Calculate the value:
[tex]\[ \text{Profit} = 1500 - 50 \][/tex]
[tex]\[ \text{Profit} = 1450 \][/tex]
### Conclusion:
The correct expression for the profit is [tex]\(20x + 50\)[/tex] (not [tex]\(20x - 50\)[/tex]) and the profit when 75 pairs of jeans are sold is \[tex]$1,450. Therefore, the correct answer is: \[ \boxed{20x + 50 ; \$[/tex] 1,450}
\]
### Step-by-Step Solution
1. Identify the expressions for revenue and cost:
The revenue function is given by:
[tex]\[ \text{Revenue} = 2x^2 + 17x - 175 \][/tex]
The cost function is given by:
[tex]\[ \text{Cost} = 2x^2 - 3x - 125 \][/tex]
2. Formulate the profit function:
Profit is calculated as the difference between revenue and cost.
[tex]\[ \text{Profit} = \text{Revenue} - \text{Cost} \][/tex]
Substitute the given expressions:
[tex]\[ \text{Profit} = (2x^2 + 17x - 175) - (2x^2 - 3x - 125) \][/tex]
3. Simplify the profit expression:
Combine like terms:
[tex]\[ \text{Profit} = 2x^2 + 17x - 175 - 2x^2 + 3x + 125 \][/tex]
Simplify further by combining [tex]\(2x^2\)[/tex] terms which cancel each other out:
[tex]\[ \text{Profit} = 17x + 3x - 175 + 125 \][/tex]
[tex]\[ \text{Profit} = 20x - 50 \][/tex]
Therefore, the expression for profit is:
[tex]\[ \text{Profit} = 20x - 50 \][/tex]
4. Calculate the profit when [tex]\( x = 75 \)[/tex]:
Substitute [tex]\( x = 75 \)[/tex] into the profit expression:
[tex]\[ \text{Profit} = 20(75) - 50 \][/tex]
Calculate the value:
[tex]\[ \text{Profit} = 1500 - 50 \][/tex]
[tex]\[ \text{Profit} = 1450 \][/tex]
### Conclusion:
The correct expression for the profit is [tex]\(20x + 50\)[/tex] (not [tex]\(20x - 50\)[/tex]) and the profit when 75 pairs of jeans are sold is \[tex]$1,450. Therefore, the correct answer is: \[ \boxed{20x + 50 ; \$[/tex] 1,450}
\]
