To determine the correct equation that represents the profit earned by the hot dog stand for [tex]\( x \)[/tex] hot dogs sold, we need to break down the problem and consider the given information step-by-step.
1. Daily Cost:
The fixed daily cost to run the hot dog stand is \[tex]$48. This is a fixed expense independent of the number of hot dogs sold.
2. Profit per Hot Dog:
The profit per hot dog sold is \$[/tex]2. This means for each hot dog sold, the profit increases by \$2.
3. Formulating the Profit Equation:
- The total earnings from selling [tex]\( x \)[/tex] hot dogs can be calculated as [tex]\( 2x \)[/tex] dollars.
- The total profit, [tex]\( y \)[/tex], is given by the total earnings minus the daily cost. Therefore,
[tex]\[ y = 2x - 48 \][/tex]
This equation indicates that the profit earned [tex]\( y \)[/tex] is a linear function of the number of hot dogs sold [tex]\( x \)[/tex], where:
- [tex]\( 2x \)[/tex] represents the revenue generated from selling [tex]\( x \)[/tex] hot dogs
- [tex]\( 48 \)[/tex] represents the fixed daily cost that needs to be subtracted from the revenue to get the profit.
4. Conclusion:
The correct equation that represents the profit earned by the hot dog stand for [tex]\( x \)[/tex] hot dogs sold is:
[tex]\[ y = 2x - 48 \][/tex]
Thus, the correct answer is:
[tex]\[ y = 2x - 48 \][/tex]
This matches the third option in the list:
[tex]\[ y = 2x - 48 \][/tex]
