When x units of a certain luxury commodity are produced, they can all be sold at a price of p thousand dollars per unit, where p= -6x +100.

A) express the revenue R(x) as a function of x.

B) How much revenue is obtained when x=15 units are produced and sold?

A manufacturer has been selling lamps at the price of $50 per lamp, and at this price consumers have been buying 3,000 lamps a month. The manufacturer wishes to raise the price and estimates that for each $1 increase in the price, 1,000 fewer lamps will be sold each month. The manufacturer can produce the lamps at a cost of $29 per lamp. Express the manufacturer’s monthly profit as a function of the price that the lamps are sold, draw the graph, and estimate the optimal selling price.

A)

The demand function is p=-6x+100

Revenue = price x quantity sold = xp = x(-6x+100) = -6x^2+100x

R= -6x^2+100x

B)

R=-6x^2+100x

At x=15, R = -6(15)^2 +100(15)

R = 150

——————————-

Revenue = (50+x)(3000-1000x)

When x=0, R=50 x 3000

When x=1, price =51 , amount sold = 2000

when x=2, price = 52, amount sold = 1000 etc.

Cost = 29×3000=87,000

Profit = Revenue-Cost = (50+x)(3000-1000x) – 87,000

Profit = 150,000-50,000x+3,000x-1,000x^2-87,000

dP/dx = -50,000+3,000-2,000x = 0

1000x = 50,000

x= 50

Optimal selling price = $50 per lamp

A)

The demand function is p=-6x+100

Revenue = price x quantity sold = xp = x(-6x+100) = -6x^2+100x

R= -6x^2+100x

B)

R=-6x^2+100x

At x=15, R = -6(15)^2 +100(15)

R = 150

——————————-

Revenue = (50+x)(3000-1000x)

When x=0, R=50 x 3000

When x=1, price =51 , amount sold = 2000

when x=2, price = 52, amount sold = 1000 etc.

Cost = 29×3000=87,000

Profit = Revenue-Cost = (50+x)(3000-1000x) – 87,000

Profit = 150,000-50,000x+3,000x-1,000x^2-87,000

dP/dx = -50,000+3,000-2,000x = 0

1000x = 50,000

x= 50

Optimal selling price = $50 per lamp

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