Find where costs and revenue meet
Break-even is the output where profit is zero. Maximum revenue, maximum profit, and equality of two machines’ costs are different questions.
At least 1667 whole units are needed to cover costs. Break-even is not maximum profit.
Set revenue equal to total cost: pq = F + vq.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
What the model means
For constant price p and variable unit cost v, every unit contributes p−v toward fixed cost. If p≤v with positive fixed cost, there is no positive finite output that covers all costs. When the quantity must be an integer, round the required units up.
Build the calculation
For two machines C₁=F₁+v₁q and C₂=F₂+v₂q, their crossover is (F₂−F₁)/(v₁−v₂). Check which is cheaper on EACH side. For demand p=a−bq, revenue is aq−bq²; maximum revenue occurs at a/(2b), while maximum profit with linear costs occurs at (a−v)/(2b), subject to feasible bounds.
Set up. Calculate. Explain.
Rebuild the starting scenario
Fixed cost: 10000 $; Selling price: 10 $/unit; Variable cost: 4 $/unit. Calculate the break-even quantity before whole-unit rounding.
Setup hint
Set revenue equal to total cost: pq = F + vq.
Open the visual solution
- Set revenue equal to total cost: pq = F + vq.
- Contribution margin = $6.00 per unit.
- Divide fixed cost by margin; round UP only when output must be whole units.
At least 1667 whole units are needed to cover costs. Break-even is not maximum profit.
Set revenue equal to total cost: pq = F + vq.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Transfer the idea to a new scenario
A product has fixed cost $22,000, price $12.50/unit, and variable cost $4.50/unit. Find break-even output.
Setup hint
Set revenue equal to total cost: pq = F + vq.
Open the visual solution
- Set revenue equal to total cost: pq = F + vq.
- Contribution margin = $8.00 per unit.
- Divide fixed cost by margin; round UP only when output must be whole units.
At least 2750 whole units are needed to cover costs. Break-even is not maximum profit.
Set revenue equal to total cost: pq = F + vq.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Why is maximum revenue not generally maximum profit?
Can you defend your setup?
Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?
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