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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]$48 each morning for the day's supply of hot dogs, buns, and mustard, but he earns \$[/tex]2 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 which equation 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:

1. Initial Cost:
The owner spends \[tex]$48 each morning for the day's supply of hot dogs, buns, and mustard. This is a fixed cost that reduces the profit. Thus, the profit starts at \(-\$[/tex]48\).

2. Profit per Hot Dog:
The owner earns \[tex]$2 profit for each hot dog sold. Hence, for each hot dog sold, the profit increases by \$[/tex]2.

3. Formulating the Equation:
- Let [tex]\( x \)[/tex] be the number of hot dogs sold.
- The total profit earned from selling [tex]\( x \)[/tex] hot dogs is [tex]\( 2x \)[/tex] dollars because each hot dog sold contributes \[tex]$2 to the profit. - Since there is an initial cost of \$[/tex]48, this amount must be subtracted from the total profit.

Putting these points together, the profit equation can be set up as follows:
[tex]\[ y = 2x - 48 \][/tex]
Where:
- [tex]\( y \)[/tex] is the total profit.
- [tex]\( 2x \)[/tex] represents the profit from selling [tex]\( x \)[/tex] hot dogs.
- The [tex]\(-48\)[/tex] represents the initial fixed cost.

Therefore, the correct equation is:
[tex]\[ y = 2x - 48 \][/tex]

So, the correct choice is the third one.

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