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=1.55x35,000=0.7x50,000=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
- Use the GeoGebra.com tool to graph the cost and revenue functions
- Identify the break-even point using the “Intersect” tool under “Points” on GeoGebra
- 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.
Break-Even Analysis and Profit Calculation in Business
In business, understanding the break-even point is crucial as it marks the level of sales at which total revenue equals total costs, resulting in neither profit nor loss. This analysis helps determine the minimum amount of sales needed to cover all expenses and start generating profit. In this essay, we will delve into the break-even analysis, profit calculation, and implications for a company's operations.
Break-Even Point and Profit Margin
The break-even point represents the level of output where revenues match total costs. Graphically, it is where the cost and revenue functions intersect. If a business's cost and revenue equations are dependent, it implies that the company's profit margins are zero until they surpass the break-even point. This signifies that the company needs to exceed this point to start making profits.
Multiple Break-Even Points
When solving a break-even analysis, encountering multiple break-even points indicates fluctuations in profit levels. These variations suggest that the company may experience periods of profitability followed by phases of losses, highlighting the importance of closely monitoring sales and costs to maintain sustainable operations.
No Break-Even Point Scenario
If a break-even analysis yields no break-even point, it suggests that either the company consistently operates at a profit or continuously sustains losses. This scenario underscores the need for a thorough review of cost structures, pricing strategies, and revenue streams to ensure financial viability.
Ensuring a Break-Even Point
To ensure a break-even point, companies should focus on optimizing operational efficiency, cost management, pricing strategies, and sales volume. By accurately assessing cost structures and setting realistic revenue targets, businesses can position themselves to achieve profitability and long-term sustainability.
Profit Calculation Problem
Let's consider an investor who earned triple the profits of what she made last year, totaling $500,000.48 for both years. To calculate how much she earned in profits each year, we can set up the equation:
Let x be the profit earned last year.
The profit earned this year is 3x.
Given that the total profit for both years is $500,000.48:
x + 3x = 500,000.48
4x = 500,000.48
x = 500,000.48 / 4
x = $125,000.12
Therefore, the investor earned $125,000.12 in profits last year and $375,000.36 this year.
Graphical Representation
A graph modeling this situation would showcase two intersecting lines representing the profit earned each year. The point of intersection would correspond to the total profit over the two years. Increasing the amount earned for both years would shift the total profit higher on the graph, while doubling the profits of a particular year would change the slope of one of the lines accordingly.
In conclusion, break-even analysis and profit calculations are vital tools for businesses to assess their financial health and make informed decisions. By understanding these concepts and leveraging them effectively, companies can enhance their profitability and navigate market challenges successfully.