The break-even points are (1 , 14) and (7 , 26), the numbers in thousands
When the company produces 1 thousand items the revenue and expenses are equal 14 thousands
When the company produces 7 thousands items the revenue and the expenses are equal 26 thousands
Step-by-step explanation:
The break-even point is the point (x, y) at which the graphs of the
revenue and cost functions intersect
For one manufacturing company, the revenue from producing
x items is given by the function y = 2x + 12, and the cost of
producing x items is given by y = -x² + 10x + 5
The numbers represent thousands
We need to find all break-even points
To solve the problem do that:
1. equate the two equations
2. Collect all terms in one side and put the other side 0
3. Factorize the quadratic to find the values of x
4. substitute the value of x in the linear equation to find the values of y
∵ y = 2x + 12
∵ y = -x² + 10x + 5
∴ 2x + 12 = -x² + 10x + 5
- Add x² to both sides
∴ x² + 2x + 12 = 10x + 5
- Subtract 10x from both sides
∴ x² - 8x + 12 = 5
- Subtract 5 from both sides
∴ x² - 8x + 7 = 0
- Factorize the left hand side
∴ (x - 1)(x - 7) = 0
- Equate each bracket by 0
∵ x - 1 = 0
- Add 1 to both sides
∴ x = 1
∵ x - 7 = 0
- Add 7 to both sides
∴ x = 7
∴ x = 1 and x = 7
Substitute the values of x in the equation y = 2x + 12 to find the
values of y
∵ y = 2x + 12
∵ x = 1
∴ y = 2(1) + 12
∴ y = 14
∵ x = 7
∴ y = 2(7) + 12
∴ y = 14 + 12
∴ y = 26
∴ y = 14 and y = 26
∴ The break-even points are (1 , 14) and (7 , 26), the numbers in
thousands
The break-even points are (1 , 14) and (7 , 26), the numbers in thousands
When the company produces 1 thousand items the revenue and expenses are equal 14 thousands
When the company produces 7 thousands items the revenue and the expenses are equal 26 thousands
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