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) = 74,000 + 80x and p(x) = 200 - x/30, 0 lessthanorequalto x lessthanorequalto 6000. (A) Fine the maximum revenue. (B) Fine 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 56 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 $. (B) The maximum profit is $ when sets are manufactured and sold for each. (C) When each set is taxed at $6, the maximum profit is $ when sets are manufactured and sale for $ each.

Explanation / Answer

We have given C(x)=74000+80x and p(x)=200-x/30,0<=x<=6000,x is quantity

A) Revenue R(x)=price *quantity

R(x)=(200-x/30)*x =200x-x^2/30

R(x) is maximized when R'(x)=0

R'(x)=200-(2x)/30=0

x/15=200

x=15*200=3000

x=3000

R(x)=200x-x^2/30

R(3000)=(200*3000)-(3000*3000)/30

=600000-300000=300000

the maximum revenue is $300000

B) Profit P(x)=Revenue-cost

P(x)=[200x-x^2/30]-[74000+80x]

P(x)=120x-x^2/30-74000

P'(x)=120-x/15=0

120=x/15

x=1800

P(x)=120x-x^2/30-74000

P(1800)=(120*1800)-((1800*1800)/30)-74000

P(1800)=$34000

p(x)=200-x/30

p(1800)=200-(1800/30)=$140

The maximum profit is $34000 when 1800 sets are manufactured and sold for $140 each

C) with $6 tax the price demand equation should be p(x)=200-x/30+6=206-x/30

p(x)=206-x/30

R(x)=x*p(x)=(206-x/30)*x=206x-x^2/30

Profit P(x)=R(x)-C(x)=206x-x^2/30-74000-80x

P(x)=126x-x^2/30-74000

P'(x)=126-x/15=0

x/15=126 implies x=15*126 =1890

x=1890

the maximum profit is P(1890)=126*(1890)-(1890*1890)/30-74000 =$45070

p(x)=206-x/30

p(1890)=206-(1890)/30=$143

when each set is taxed $6 the maximum profit is $45070 when 1890 sets are manufactured and sold for $143 each

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