A large company in the communication and publishing industry has quantified the relationship between the price of one of its products and the demand for this product as Price=150−0.02×Demand for an annual printing of this particular product. The fixed costs per year​ (i. e., per ​printing)=​$55,000 and the variable cost per unit=​$50. What is the maximum profit that can be​ achieved? What is the unit price at this point of optimal​ demand? Demand is not expected to be more than 3,000 units per year.

Explanation:

From the information given:

The first thing is to calculate the total cost:

The total cost = fixed cost  + Variable cost

= 55000 + 50Q

The total revenue TR = Price (P) × Demand (Q)

= (150 - 0.02Q) × Q

= 150Q - 0.02Q²

The marginal revenue MR =

MR = 150 - 0.04Q

The marginal cost

MC = 50

Now, the profit can be accomplished at the point when marginal revenue is equivalent to the marginal cost.

Then;

MR = MC

150 - 0.04Q = 50

-0.04Q = -150 + 50

-0.04 Q = -100

Q = 100/0.04

Q = 2500

Replacing the value of Q into P =150 - 0.02Q

P = 150 - 0.02(2500)

P = 150 - 50

P = $100

So, Profit = Total revenue - Total cost

Profit = (150Q - 0.02Q²) - ( 55000 + 50Q)

Profit = (150Q - 0.02Q² - 55000 - 50Q

Profit = 100Q - 0.02Q² - 55000

Profit = 100(2500) - 0.02(2500)² - 55000

Profit = 250000 - 125000 - 55000

Profit = $70000

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