A small plant manufactures riding lawn mowers. The plant has fixed costs​ (leases, insurance, and so​ on) of ​$24000 per day and variable costs​ (labor, materials, and so​ on) of ​$1400 per unit produced. The mowers are sold for ​$1700 each. The cost and revenue equations are shown below where x is the total number of mowers produced and sold each​ day, and the daily costs and revenue are in dollars. Complete parts​ (A) and​ (B).

 A small plant manufactures riding lawn mowers. The plant has fixed costs​ (leases, insurance, and so​ on) of ​$24000 per day and variable costs​ (labor, materials, and so​ on) of ​$1400 per unit produced. The mowers are sold for ​$1700 each. The cost and revenue equations are shown below where x is the total number of mowers produced  and sold each​ day, and the daily costs and revenue are in dollars. Complete parts​ (A) and​ (B).

 

 (A) How many units must be manufactured and sold each day for the company to break​ even?

 

​(B) Graph both equations in the same coordinate system and show the​ break-even point. Interpret the regions between the lines to the left and to the right of the​ break-even point.

One way to graph the equations is to find two points on each line.
Since the​ break-even point is on the line of both​ equations, it is a good point to find. Substitute 150
 
for x in either the cost equation or the revenue​ equation, and calculate the corresponding​ y-coordinate. This example uses the revenue equation since it has fewer terms.
 

 

Plot the three points and draw lines from each​ y-intercept through the​ break-even point. A graph of both equations in the same coordinate system showing the​ break-even point is shown to the right.

 

To the left of the​ break-even point, the line of the cost equation is above the line of the revenue equation. Since the cost is greater than the revenue for all​ x-values less than the​ x-coordinate of the​ break-even point, the region between the lines to the left of the​ break-even point represents loss.
To the right of the​ break-even point, the line of the revenue equation is above the line of the cost equation. Since the revenue is greater than the cost for all​ x-values greater than the​ x-coordinate of the​ break-even point, the region between the lines to the right of the​ break-even point represents profit.