A company manufactures and sells x transistor radios per week. The weekly cost and price demand equations are C(x)=5000+2x and p=10-.001x where x can be no more than 10000 units. The government has decided to tax the company $2 for each radio produced. Taking into account this additional cost, how many radio should the company manufacture each week in order to maximize its weekly profit? What is the maximum weekly profit? How much should the company charge for the radios to realize the maximum weekly profit?
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The company manufactures and sells x transistor radios per week. The cost of producing x radios is given by C(x)=5000+2x and the price demand equation is P = 10 - .001x where x can be no more than 10000 units.
An additional tax of $2 for each radio is introduced by the government, this changes the cost function to C(x) = 5000 + 2x + 2x = 5000 + 4x
If the profit made by the company is maximized when x radios are manufactured, the total cost incurred is 5000 + 4x and the revenue is x*(10 - .001*x). The profit made is P(x) = 10x - 0.001x^2 - 5000 - 4x
= -0.001x^2 + 6x - 5000
Solve P'(x) = 0 for x
-2*0.001x + 6 = 0
=> x = 3000
The maximum profit is $4000. The radios should be priced at $7
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