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A carpenter in Brisbane is building two types of wooden shelves: standard shelves cost $24 to make and sell for $40, while deluxe shelves cost $36 to make and sell for $60. The carpenter has a weekly budget of $1200 for materials and can build at most 40 shelves per week.
Let x be the number of standard shelves and y be the number of deluxe shelves. The constraints are 24x + 36y ≤ 1200 and x + y ≤ 40.
The carpenter wants to know how many of each shelf type must be made to break even exactly on total cost and total revenue, assuming all shelves are sold. Solve the system of equations 24x + 36y = 1200 and 40x + 60y = 1200 exactly and find the value of x (number of standard shelves).
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Worked solution
- The carpenter's total cost to make x standard shelves and y deluxe shelves is 24x + 36y. The total revenue from selling them is 40x + 60y. To break even exactly, total cost must equal total revenue, so 24x + 36y = 40x + 60y.
- Subtract 24x and 36y from both sides to form a homogeneous equation: 0 = 16x + 24y.
- Simplify 16x + 24y = 0 by dividing all terms by 8: 2x + 3y = 0.
- Solve 2x + 3y = 0 for x: 2x = -3y, so x = -3y/2.
- Substitute x = -3y/2 into the materials budget constraint: 24x + 36y ≤ 1200. Use equality for break-even point: 24(-3y/2) + 36y = 1200.
- Calculate: 24 × (-3y/2) = -36y. So -36y + 36y = 0 = 1200. This gives 0 = 1200, which is a contradiction.
- The contradiction means that the break-even point cannot occur under the cost and revenue equations as derived. Re-express the break-even condition correctly: total cost = total revenue, so 24x + 36y = 40x + 60y.
- Bring all terms to one side: 24x + 36y - 40x - 60y = 0 → -16x - 24y = 0.
- Simplify: divide by -8 → 2x + 3y = 0. This again gives x = -3y/2.
- Since x and y represent numbers of shelves, they must be non-negative integers. But x = -3y/2 implies x is negative if y > 0. The only solution is x = 0, y = 0.
- Check: if x = 0, y = 0, cost = 0, revenue = 0 → break-even. But this means no shelves are made. The question asks for a non-trivial break-even point where shelves are made and sold.
- Re-express the correct break-even condition: total cost = total revenue, so 24x + 36y = 40x + 60y. Rearranged: 0 = 16x + 24y. This implies 2x + 3y = 0. Only solution with x ≥ 0, y ≥ 0 is x = 0, y = 0. Therefore, no positive integer solution exists for break-even under these cost and revenue structures.
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