To determine the equation that represents [tex]\( y \)[/tex], the profit earned by the hot dog stand for [tex]\( x \)[/tex] hot dogs sold, let's break down the problem step-by-step.
### Step-by-Step Solution:
1. Understand the Costs and Profits:
- The daily cost for the hot dog stand is [tex]$48. This is a fixed cost that the owner incurs regardless of the number of hot dogs sold.
- The profit for each hot dog sold is $[/tex]2.
2. Set Up the Profit Equation:
- The profit equation generally takes the form of [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the profit per unit (slope), and [tex]\( b \)[/tex] is the initial cost (y-intercept).
3. Define the Variables:
- Let [tex]\( x \)[/tex] represent the number of hot dogs sold.
- Let [tex]\( y \)[/tex] represent the total profit earned.
4. Calculate the Total Profit:
- The total earnings from selling [tex]\( x \)[/tex] hot dogs is [tex]\( 2x \)[/tex] dollars (since the profit per hot dog is [tex]$2).
- However, there is a fixed daily cost of $[/tex]48, so this amount will be subtracted from the total earnings to find the profit.
5. Formulate the Equation:
- The profit [tex]\( y \)[/tex] will be the earnings from selling hot dogs minus the daily cost:
[tex]\[
y = 2x - 48
\][/tex]
6. Verify the Equation:
- If zero hot dogs are sold ( [tex]\( x = 0 \)[/tex] ), the profit [tex]\( y \)[/tex] will be:
[tex]\[
y = 2(0) - 48 = -48
\][/tex]
This means the owner incurs a loss of $48, which is the cost of the daily supplies — consistent with the given problem.
### Conclusion:
The equation that accurately represents [tex]\( y \)[/tex], the profit earned by the hot dog stand for [tex]\( x \)[/tex] hot dogs sold, is:
[tex]\[
y = 2x - 48
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{y = 2x - 48}
\][/tex]
