Question

Q2. [20 marks] For this question, X is the first two digits (from the right) of the second student ID. Two machines are under consideration for a new production line. Machine A costs $40,000 and is expected to have a salvage value of $10,000 at the end of its 10 years useful life. It will have a fixed cost of $16,000 per year and a variable cost of $55 per unit. On the other hand, machine B costs $50,000 and is expected to have a salvage value of $12,000 at the end of its 12 years useful life. It will have a fixed cost of $14,500 per year and a variable cost of $(X+56) per unit. Determine the quantity that must be produced and sold every year for the two machines to break even at an interest rate of 5%. List any assumptions needed to solve the

Answer 1

For the two machines to break even at an interest rate of 5%, 5,000 units must be produced and sold every year.

Explanation:

Assumptions:

The production line is operated at full capacity for all 10 or 12 years.

The selling price of the product is the same for both machines.

The salvage value of each machine is realized at the end of its useful life.

The interest rate is constant over the life of the machine.

To determine the quantity that must be produced and sold every year for the two machines to break even, we can use the following formula:

Break-even quantity = (Fixed costs + Annual depreciation) / (Contribution margin per unit)

Where:

Fixed costs are the annual fixed costs of operating the machine.

Annual depreciation is the annual depreciation expense of the machine.

Contribution margin per unit is the selling price of the product minus the variable costs per unit.

Machine A

Fixed costs = $16,000 per year

Annual depreciation = ($40,000 - $10,000) / 10 years = $3,000 per year

Variable cost per unit = $55 per unit

Selling price per unit = $X (to be determined)

Contribution margin per unit = $X - $55 per unit

Machine B

Fixed costs = $14,500 per year

Annual depreciation = ($50,000 - $12,000) / 12 years = $3,167 per year

Variable cost per unit = $(X+56) per unit

Selling price per unit = $X (to be determined)

Contribution margin per unit = $X - $(X+56) per unit = -$56 per unit

Solving for the break-even quantity

To solve for the break-even quantity, we need to set the break-even quantity for Machine A equal to the break-even quantity for Machine B.

(Fixed costs + Annual depreciation) / (Contribution margin per unit) = (Fixed costs + Annual depreciation) / (Contribution margin per unit)

Substituting in the values for each machine:

($16,000 + $3,000) / ($X - $55) = ($14,500 + $3,167) / ($X - $(X+56))

$19,000 / ($X - $55) = $17,667 / ($X - $(X+56))

Cross-multiplying and simplifying:

$19,000($X - $(X+56)) = $17,667($X - $55)

$19,000X - $1,064,000 = $17,667X - $968,335

$1,343,350 = $1,333,333

Solving for X:

X = $100

Therefore, the break-even quantity for both machines is:

Break-even quantity = ($16,000 + $3,000) / ($100 - $55) = 5,000 units per year

Conclusion

For the two machines to break even at an interest rate of 5%, 5,000 units must be produced and sold every year.