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Given the cost function C(x)=0.85x+35,000?(?)=0.85?+35,000 and the revenue function R(x)=1.55x,?(?)=1.55?, the break-even point is (50,000,77,500) and the profit function is P(x)=0.7x−35,000.
Solution
Write the system of equations using y? to replace function notation.
y=0.85x+35,000y=1.55x?=0.85?+35,000?=1.55?
Substitute the expression 0.85x+35,0000.85?+35,000 from the first equation into the second equation and solve for x.?.
0.85x+35,000=1.55×35,000=0.7×50,000=x0.85?+35,000=1.55?35,000=0.7?50,000=?
Then, we substitute x=50,000?=50,000 into either the cost function or the revenue function.
1.55(50,000)=77,5001.55(50,000)=77,500
The break-even point is (50,000,77,500).(50,000,77,500).
The profit function is found using the formula P(x)=R(x)−C(x).?(?)=?(?)−?(?).
P(x)=1.55x−(0.85x+35,000) =0.7x−35,000?(?)=1.55?−(0.85?+35,000) =0.7?−35,000
The profit function is P(x)=0.7x−35,000.
The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units.
Assignment
1. Use the GeoGebra.com tool to graph the cost and revenue functions
2. Identify the break-even point using the “Intersect” tool under “Points” on GeoGebra
3. Save your GeoGebra work as a .pdf file for submission
Discuss the part of the graph that represents the profit.
Discuss how you found the break-even point on the graph.
If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.
If you are solving a break-even analysis and get more than one break-even point, explain what this signifies for the company?
If you are solving a break-even analysis and there is no break-even point, explain what this means for the company.
How should they ensure there is a break-even point?
Solve the following problem: An investor earned triple the profits of what she earned last year. If she made $500,000.48 total for both years, how much did she earn in profits each year?
Write an analysis of your solution to this problem.
Describe the graph that could model this situation.
Discuss how your answer would be affected if:
The amount earned for both years was increased.
The investor only earned double the profits of what she earned last year.
Discussion may include: On the graph, the region of profit will be where the revenue function values are higher than the cost function values. The break-even point will be the intersection points of the two graphs. Having more than one break-even point means that there are alternations of profit and loss. Dependent equations would mean they are the same curve and no profit exists. If there is no break-even point, then either there was continuous profit or there was continuous loss.
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