Finding a Break-Even Point

1. ⓐ The cost function in the sum of the fixed cost, $125,000, and the variable cost, $120 per helmet.
   $$C\left(x\right)=120x+250,000$$
2. ⓑ The revenue function is the total revenue from the sale of $x$ helmets, $R\left(x\right)=140x.$
3. ⓒ The break-even point is the point of intersection of the graph of the cost and revenue functions. To find the x-coordinate of the coordinate pair of the point of intersection, set the two equations equal, and solve for $x.$
   $$\begin{array}{l}C(x)=R(x)\hfill \\ 250,000+120x=140x\hfill \\ 250,000=20x\hfill \\ 12,500=x\hfill \\ x=12,500\hfill \end{array}$$

   To find $y,$ evaluate either the revenue or the cost function at 12,500.

   $$\begin{array}{l}R(x)=140(12,500)\hfill \\ =\$1,750,000\hfill \end{array}$$

The break-even point is $\left(\mathrm{12,500,}\mathrm{1,750,000}\right).$