1. ## Application problem

The wholesale price for chicken in the United States fell from 25 cents per pound to 14 cents per pound while per capita chicken consumption rose from 22 pounds per year to 27.5 pounds per year. Assuming that the demand for chicken depends linearly on the price, what wholesale price for chicken maximizes revenues for poultry farmers and what does that revenue amount to?

I know this is a simple problem but I have no idea as to what equation I have to write to find the maximum.

2. Originally Posted by Kenneth
The wholesale price for chicken in the United States fell from 25 cents per pound to 14 cents per pound while per capita chicken consumption rose from 22 pounds per year to 27.5 pounds per year. Assuming that the demand for chicken depends linearly on the price, what wholesale price for chicken maximizes revenues for poultry farmers and what does that revenue amount to?

I know this is a simple problem but I have no idea as to what equation I have to write to find the maximum.
Hi, Kenneth,

I've got a funny result and I believe that it is wrong, but I send it to you nevertheless:

The revenue is calculated by weight * price per pound. The result is the money the farmer gets from every chicken eating American.

The weight went up by 5.5 pounds when the price goes down by 11 cts. Thus: Decreasing the price by 1 ct. would increase the weight by 0.5 pounds.

Therefore:

Let x be the number of cents the price per pound decreases.
r is the revenue
w is the weight
p is the price per pound:

r = w * p

If the price changes you get:

r(x) = (22+0.5x)(25-x) = -0.5x² - 9.5x + 550

The graph of this function is a parabola opening downward. The vertex of the parabola is the maximum of the function. The x-coordinate of the vertex is (-b)/(2a). The maximum will be reached if x = (-(-9.5))/(2*(-0.5)) = -9.5

You have to increase the price to 34.5 cts per pound. The consumption will go down to 17.5 pounds per person and then the revenue will be 595.125 cts per chicken eating person.

Well this result is confusing me a lot, but I cann't detect my mistake. Maybe you are more successfull. Good luck!

EB

3. Hello, Kenneth!

My approach differs slightly from EB's, but we agree on the answer.

The wholesale price for chicken in the United States fell from 25¢/lb to 14¢/lb
while per capita chicken consumption rose from 22 lb/yr to 27.5 lb/yr.
Assuming that the demand for chicken depends linearly on the price,
what wholesale price for chicken maximizes revenues for poultry farmers
and what does that revenue amount to?

Let P = price, D = demand.

We have two points: .(P,D) = (25, 22), (14, 27.5)

The slope is: .m .= .(27.5 - 22)/(14 - 25) .= .

The function is: .D - 22 .= .-½(P - 25) . . D .= .-½P + 34.5

Revenue .= .(Price) x (Demand)

. . R .= .P(-½P + 34.5) .= .-½(P² - 69P)

To maximize Revenue, solve: .R' = 0

. . R' .= .-½(2P - 69) .= .0 . . P = 34.5

Therefore, charging 34.5¢/lb will maximize revenue.

The maximum revenue is: .-½[34.5² - 69(34.5)] .= .595.125¢ . .\$5.95

4. Thankyou. Both of your solutions were a great help.