Question

The demand for gourmet chocolate cakes sold by a local bakery is estimated to follow an exponential distribution with a mean of 130 cakes f(x)={1/130e⁻ˣ/¹³⁰, x≥0 otherwise.

Each chocolate cake costs the owner of the bakery $20 to bake and are sold for $40 per cake. If a customer arrives in the bakery to buy a chocolate cake, and none is available, the owner gives the customer a discount voucher costing $20 as a goodwill gesture. Any gourmet chocolate cakes left over at the end of the week can be sold for $10 per cake.
How many gourmet chocolate cakes should the bakery bake for sale?

Answer 1

To determine how many gourmet chocolate cakes the bakery should bake for sale, calculate the expected demand, revenue, cost, and then find the optimal quantity of cakes.

Step-by-step explanation:

To determine how many gourmet chocolate cakes the bakery should bake for sale, we need to consider the cost and revenue associated with baking cakes. Let's break down the calculation step by step:

1. Calculate the expected demand for cakes:

• Since the demand follows an exponential distribution, we can use the mean as the expected value. The mean demand is 130 cakes.

Calculate the expected revenue:

• For each cake sold, the bakery earns $40.
• If a cake is sold at a discount, the bakery incurs a cost of $20.
• If a cake is sold at the end of the week, the bakery earns $10.

Calculate the expected cost:

• Each cake costs $20 to bake.

Determine the optimal quantity of cakes to bake:

• The bakery should bake enough cakes to meet the expected demand while minimizing costs and maximizing revenue.
• This can be done by comparing the expected revenue and the expected cost for different quantities of cakes.
• Find the quantity that maximizes the difference between the expected revenue and the expected cost.

By following these steps, the bakery can determine the optimal quantity of gourmet chocolate cakes to bake for sale.