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The profit earned by a hot dog stand is a linear function of the number of hot dogs sold. It costs the owner [tex]$\$[/tex]48[tex]$ each morning for the day's supply of hot dogs, buns, and mustard, but he earns $[/tex]\[tex]$2$[/tex] profit for each hot dog sold.

Which equation represents [tex]$y$[/tex], the profit earned by the hot dog stand for [tex]$x$[/tex] hot dogs sold?

A. [tex]$y = 48x - 2$[/tex]
B. [tex]$y = 48x + 2$[/tex]
C. [tex]$y = 2x - 48$[/tex]
D. [tex]$y = 2x + 48$[/tex]



Answer :

GinnyAnswer
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.

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