To determine the equation that represents the profit earned by the hot dog stand for [tex]\( x \)[/tex] hot dogs sold, we need to analyze the given information step-by-step.
1. Identify the variables:
- [tex]\( x \)[/tex] is the number of hot dogs sold.
- [tex]\( y \)[/tex] is the profit earned.
2. Initial costs and profits:
- The owner incurs a fixed cost of \[tex]$48 each morning for supplies.
- The owner earns a profit of $[/tex]2 for each hot dog sold.
3. Formulating the profit function:
- The total earnings for selling [tex]\( x \)[/tex] hot dogs is [tex]\( 2x \)[/tex] because each hot dog contributes \[tex]$2 to the profit.
- The total cost for the day is a fixed \(\$[/tex]48\).
4. Calculating the net profit:
- Net profit [tex]\( y \)[/tex] can be calculated by subtracting the total daily cost from the total earnings.
- Therefore, the profit function is given by:
[tex]\[
y = 2x - 48
\][/tex]
5. Selecting the correct equation from the options provided:
- Option 1: [tex]\( y = 48x - 2 \)[/tex] — This is incorrect because it misrepresents the coefficients and operation structure.
- Option 2: [tex]\( y = 48x + 2 \)[/tex] — This is incorrect for the same reason as above.
- Option 3: [tex]\( y = 2x - 48 \)[/tex] — This matches our derived equation.
- Option 4: [tex]\( y = 2x + 48 \)[/tex] — This is incorrect because it does not account for the initial cost being subtracted.
Therefore, the equation that represents the profit earned by the hot dog stand for [tex]\( x \)[/tex] hot dogs sold is:
[tex]\[ y = 2x - 48 \][/tex]
And this corresponds to option number 3.
