3. A beekeeper plans to sell honey at the Lehigh University Thursday local farmers' market. The beekeeper makesa net profit of $40 per kilogram on honey she sells, but she incurs a net loss of $10 per kilogram on any honey shebrings to the market that she doesn't sell. The (random) demanded quantity X for honey at the farmers' market(in kilograms) is uniform on (0,5). Let k be the amount of honey (in kilograms) that the beekeeper brings. (Tobe clear, if the demand meets or exceeds k, she sells all of the honey she brings. If the demand falls short ofk, then she sells an amount of honey equal to the quantity demanded, and she incurs a loss on the amount shebrings beyond the demand.) What should k be (that is, how much honey should she bring) for the beekeeper tomaximize her expected earnings from the farmers' market?

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Asked Nov 21, 2019
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3.

3. A beekeeper plans to sell honey at the Lehigh University Thursday local farmers' market. The beekeeper makes
a net profit of $40 per kilogram on honey she sells, but she incurs a net loss of $10 per kilogram on any honey she
brings to the market that she doesn't sell. The (random) demanded quantity X for honey at the farmers' market
(in kilograms) is uniform on (0,5). Let k be the amount of honey (in kilograms) that the beekeeper brings. (To
be clear, if the demand meets or exceeds k, she sells all of the honey she brings. If the demand falls short of
k, then she sells an amount of honey equal to the quantity demanded, and she incurs a loss on the amount she
brings beyond the demand.) What should k be (that is, how much honey should she bring) for the beekeeper to
maximize her expected earnings from the farmers' market?
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3. A beekeeper plans to sell honey at the Lehigh University Thursday local farmers' market. The beekeeper makes a net profit of $40 per kilogram on honey she sells, but she incurs a net loss of $10 per kilogram on any honey she brings to the market that she doesn't sell. The (random) demanded quantity X for honey at the farmers' market (in kilograms) is uniform on (0,5). Let k be the amount of honey (in kilograms) that the beekeeper brings. (To be clear, if the demand meets or exceeds k, she sells all of the honey she brings. If the demand falls short of k, then she sells an amount of honey equal to the quantity demanded, and she incurs a loss on the amount she brings beyond the demand.) What should k be (that is, how much honey should she bring) for the beekeeper to maximize her expected earnings from the farmers' market?

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Expert Answer

Step 1

The distribution of demand follows uniform (0, 5)

1/5
5
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1/5 5

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Step 2

There are two cases: k>d and kd

For k>d case:
Profit 40d-(k- d)10
P 50d 10k
For kd case:
Profit 40k
40k
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For k>d case: Profit 40d-(k- d)10 P 50d 10k For kd case: Profit 40k 40k

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Step 3
P (k>d) Area of shaded region
-(k
5
P(k d) Area of unshaded region
-(5-k)
5
The expected profit is calculated as follows:
E (Profit)- P(k> d)P + P(k s d)P
5 k
k
(50d-10k)
( 40k)
5
Here 0sdsk& uniform
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P (k>d) Area of shaded region -(k 5 P(k d) Area of unshaded region -(5-k) 5 The expected profit is calculated as follows: E (Profit)- P(k> d)P + P(k s d)P 5 k k (50d-10k) ( 40k) 5 Here 0sdsk& uniform

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