Question

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?

Answer 1

Answer:

Part (A)

• 1. Maximum revenue: $450,000

Part (B)

• 2. Maximum protit: $192,500

• 3. Production level: 2,300 television sets

• 4. Price: $185 per television set

Part (C)

• 5. Number of sets: 2,260 television sets.

• 6. Maximum profit: $183,800

• 7. Price: $187 per television set.

Explanation:

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

Cost:

      $C(x)=72,000+70x$

Price-demand:

     

      $p(x)=300-\dfrac{x}{20}$

Domain:

        $0\leq x\leq 6000$

1. Part (A) Find the maximum revenue

Revenue = price × quantity

Revenue = R(x)

           $R(x)=\bigg(300-\dfrac{x}{20}\bigg)\cdot x$

Simplify

      $R(x)=300x-\dfrac{x^2}{20}$

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

         $R'(x)=300-\dfrac{x}{10}$

Solve for R'(x)=0

      $300-\dfrac{x}{10}=0$

       $3000-x=0\\\\x=3000$

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)

       $Profit(x)=300x-\dfrac{x^2}{20}-\big(72,000+70x\big)$

       $Profit(x)=230x-\dfrac{x^2}{20}-72,000\\\\\\Profit(x)=-\dfrac{x^2}{20}+230x-72,000$

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 = 0

• x = 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/20

• p(2,300) = 300 - 2,300 / 20

• p(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 - 72,000

• Profit(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 = 0

• x = 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) - 72,000

• Profit (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 - 2,260/20

• p(2,260) = 187.

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