Let's break down the problem:
1. Identify the costs and selling price:
- The cost to make each button is \[tex]$0.75.
- The selling price for each button is \$[/tex]2.
2. Calculate the profit per button:
- The profit per button is the difference between the selling price and the cost to make:
[tex]\[
\text{Profit per button} = \text{Selling price} - \text{Cost to make} = \$2 - \$0.75 = \$1.25
\][/tex]
3. Set up the equation for the desired profit:
- Let [tex]\( b \)[/tex] be the number of buttons sold.
- The total profit from selling [tex]\( b \)[/tex] buttons can be calculated by multiplying the number of buttons by the profit per button:
[tex]\[
\text{Total profit} = b \times \$1.25
\][/tex]
4. Substitute in the desired profit:
- The desired profit is \[tex]$500. So we set up the equation:
\[
\$[/tex]500 = b \times \[tex]$1.25
\]
5. Solve for \( b \):
- To find \( b \), we divide both sides of the equation by \$[/tex]1.25:
[tex]\[
b = \frac{\$500}{\$1.25} = 400
\][/tex]
Therefore, 400 buttons must be sold to make a profit of \[tex]$500.
Now, let's match this to the given answer choices:
- Option A: \$[/tex]500 = \[tex]$2b + \$[/tex]0.75b
This represents the combined cost and revenue, which is incorrect for finding the profit.
- Option B: \[tex]$500 + \$[/tex]2b = \[tex]$0.75b
This does not correctly represent the relationship between profit and buttons sold.
- Option C: \$[/tex]500 = \[tex]$2b - \$[/tex]0.75b
This simplifies to our correct equation:
[tex]\[\$500 = b \times \$1.25\][/tex]
- Option D: \[tex]$500 - \$[/tex]0.75b = \$2b
This does not correctly balance the profit calculation.
The correct answer is Option C.
