A company manufactures and sells x television sets per month. The monthly cost a

Question

A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are C(x)= 73,000 + 80x and p(x)-300- 30 0sx 9000 (A) Find the maximum revenue (B) Find the maximum profit, the production level that will realize the maximum profit, and the price the company should charge for each television set. (C) If the government decides to tax the company $5 for each set it produces, how many sets should the company manufacture each month to maximize its profit? What is the maximum profit? What should the company charge for each set? (A) The maximum revenue is $.. (Type an integer or a decimal.) sets are manufactured and sold for S (B) The maximum profit is (Type integers or decimals.) when each. (C) When each set is taxed at $6, the maximum profit is S when sets are manufactured and sold for $ each (Type integers or decimals.)

Explanation / Answer

(A) p(x)=300?(x/30?),

revenue R(x)=p*x

revenue R(x)=300x -(x2/30)

for maximum revenue dR/dx =0 ,

=>300-(2x/30)=0

=>x/10=150

=>x=1500

maximum revenue = R(1500)=300*1500 -(15002/30)

maximum revenue = R(1500)=$375000/-

(B)

profit =revenue -cost

profit P(x)=300x -(x2/30)-73000-80x

profit P(x)=220x -(x2/30)-73000

for maximum cost dP/dx =0

220 -(2x/30)=0

x=220*30/2

x=3300

p(3300)=300?(3300/30?)=300-110=$190

profit P(3300)=220*3300 -(33002/30)-73000 =290000

The maximum profit is 290000$ when 3300 sets are manufactured and sold for 190$ each

(c)

profit =revenue -cost -tax

profit P(x)=300x -(x2/30)-73000-80x-5x

profit P(x)=215x -(x2/30)-73000

for maximum cost dP/dx =0

215-(2x/30)=0

x=215*15

x=3225

p(3225)=300?(3225/30?)=192.5

profit P(3225)=215*3225 -(32252/20)-73000

profit P(1850)=100343.75$

When each set is taxed at ?$55, the maximum profit is 100343.75$ when 3225 sets are manufactured and sold for 192.5$ each.

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