A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are C(x)equals72 comma 000 plus 70 x and p (x )equals 300 minus StartFraction x Over 20 EndFraction , 0less than or equalsxless than or equals6000. (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 $4 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?

Part (A)

1. Maximum revenue: $450,000

Part (B)

2. Maximum protit: $192,5003. Production level: 2,300 television sets4. Price: $185 per television set

Part (C)

5. Number of sets: 2,260 television sets.6. Maximum profit: $183,8007. Price: $187 per television set.

Explanation:

0. Write the monthly cost and price-demand equations correctly:

Cost:

     

Price-demand:

     

     

Domain:

       

1. Part (A) Find the maximum revenue

Revenue = price × quantity

Revenue = R(x)

           

Simplify

     

A local maximum (or minimum) is reached when the first derivative, R'(x), equals 0.

         

Solve for R'(x)=0

     

       

Is this a maximum or a minimum? Since the coefficient of the quadratic term of R(x) is negative, it is a parabola that opens downward, meaning that its vertex is a maximum.

Hence, the maximum revenue is obtained when the production level is 3,000 units.

And it is calculated by subsituting x = 3,000 in the equation for R(x):

R(3,000) = 300(3,000) - (3000)² / 20 = $450,000

Hence, the maximum revenue is $450,000

2. Part (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.

i) Profit(x) = Revenue(x) - Cost(x)

Profit (x) = R(x) - C(x)

       

       

ii) Find the first derivative and equal to 0 (it will be a maximum because the quadratic function is a parabola that opens downward)

Profit' (x) = -x/10 + 230 -x/10 + 230 = 0-x + 2,300 = 0x = 2,300

Thus, the production level that will realize the maximum profit is 2,300 units.

iii) Find the maximum profit.

You must substitute x = 2,300 into the equation for the profit:

Profit(2,300) = - (2,300)²/20 + 230(2,300) - 72,000 = 192,500

Hence, the maximum profit is $192,500

iv) Find the price the company should charge for each television set:

Use the price-demand equation:

p(x) = 300 - x/20p(2,300) = 300 - 2,300 / 20p(2,300) = 185

Therefore, the company should charge a price os $185 for every television set.

3. Part (C) If the government decides to tax the company $4 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?

i) Now you must subtract the $4  tax for each television set, this is 4x from the profit equation.

The new profit equation will be:

Profit(x) = -x² / 20 + 230x - 4x - 72,000Profit(x) = -x² / 20 + 226x - 72,000

ii) Find the first derivative and make it equal to 0:

Profit'(x) = -x/10 + 226 = 0-x/10 + 226 = 0-x + 2,260 = 0x = 2,260

Then, the new maximum profit is reached when the production level is 2,260 units.

iii) Find the maximum profit by substituting x = 2,260 into the profit equation:

Profit (2,260) = -(2,260)² / 20 + 226(2,260) - 72,000Profit (2,260) = 183,800

Hence, the maximum profit, if the government decides to tax the company $4 for each set it produces would be $183,800

iv) Find the price the company should charge for each set.

Substitute the number of units, 2,260, into the equation for the price:

p(2,260) = 300 - 2,260/20p(2,260) = 187.

That is, the company should charge $187 per television set.

The answer is 1 because you have to flip the sign when dividing by negative number and 3

Neither is the answer