High School

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Suppose that a bakery specializes in chocolate cakes. Each cake retails at $20, and it costs $10 to prepare. Cakes cannot be sold after one week, and they have negligible salvage value. The estimated weekly demand for cakes is as follows:

- 15 cakes in 5% of the weeks
- 16 cakes in 30% of the weeks
- 17 cakes in 25% of the weeks
- 18 cakes in 20% of the weeks
- 19 cakes in 10% of the weeks
- 20 cakes in 10% of the weeks

How many cakes should the bakery prepare each week?

Answer :

The expected weekly profit would be $180

Expected weekly demand = ∑ (Demand × Probability)

= (15 × 0.05) + (16 × 0.30) + (17 × 0.25) + (18 × 0.20) + (19 × 0.10) + (20 × 0.10)

= 0.75 + 4.8 + 4.25 + 3.6 + 1.9 + 2

= 17.5 cakes

By preparing 18 cakes, the bakery can minimize potential losses due to overproduction or underproduction.

Revenue from selling x cakes = $20x

Cost of preparing x cakes = $10x

The bakery's profit function is:

Profit = Revenue - Cost

= $20x - $10x

= $10x

Since the optimal production quantity is 18 cakes, the expected weekly profit would be:

Profit = $10 × 18

= $180

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