Determine the break-even quantity of XYZ Manufacturing Co., given the following data: total fixed cost, $1200; variable cost per unit, $2; total revenue for selling q units, yTR =... Determine the break-even quantity of XYZ Manufacturing Co., given the following data: total fixed cost, $1200; variable cost per unit, $2; total revenue for selling q units, yTR = 100√q.
Understand the Problem
The question is asking to determine the break-even quantity for XYZ Manufacturing Co. using given cost and revenue equations. It involves setting the total revenue equal to the total cost and solving for the quantity (q) output that leads to neither profit nor loss.
Answer
The break-even quantities are \( q = 400 \) and \( q = 900 \).
Answer for screen readers
The break-even quantities for XYZ Manufacturing Co. are ( q = 400 ) and ( q = 900 ).
Steps to Solve
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Identify the Revenue and Cost Equations
The total revenue ($y_{TR}$) and total cost ($y_{TC}$) equations are given as:
$$ y_{TR} = 100\sqrt{q} $$ $$ y_{TC} = 2q + 1200 $$
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Set Revenue Equal to Cost
To determine the break-even quantity, set total revenue equal to total cost:
$$ 100\sqrt{q} = 2q + 1200 $$
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Simplify the Equation
Divide both sides by 2 to simplify:
$$ 50\sqrt{q} = q + 600 $$
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Rearrange and Square Both Sides
Rearranging gives:
$$ 50\sqrt{q} - q - 600 = 0 $$
Now square both sides:
$$ (50\sqrt{q})^2 = (q + 600)^2 $$
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Expand the Squared Equation
Expanding both sides results in:
$$ 2500q = q^2 + 1200q + 360000 $$
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Rearrange to Form a Quadratic Equation
Rearranging gives:
$$ 0 = q^2 - 1300q + 360000 $$
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Apply the Quadratic Formula
Use the quadratic formula ( q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) where ( a = 1, b = -1300, c = 360000 ):
$$ q = \frac{1300 \pm \sqrt{(-1300)^2 - 4 \cdot 1 \cdot 360000}}{2 \cdot 1} $$
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Calculate the Discriminant and Solve
Calculate:
$$ q = \frac{1300 \pm \sqrt{1690000 - 1440000}}{2} $$ $$ q = \frac{1300 \pm \sqrt{250000}}{2} $$
This gives:
$$ q = \frac{1300 \pm 500}{2} $$
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Find the Values for q
This results in two potential quantities:
$$ q = \frac{1800}{2} = 900 $$ $$ q = \frac{800}{2} = 400 $$
The break-even quantities for XYZ Manufacturing Co. are ( q = 400 ) and ( q = 900 ).
More Information
The break-even point indicates the output level where total revenue equals total costs, meaning the company neither makes a profit nor incurs a loss. It's important to note that producing beyond this level does not guarantee profitability, as variable costs will alter the net results.
Tips
- Mistaking the signs when rearranging the equation; always keep track of positive and negative signs.
- Forgetting to square both sides correctly, leading to incorrect terms in the expanded equation.
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