Suppose that in a monopoly market the total cost per week of producing a high tech product is given by C(x) =3600+100x+2x^2. Suppose further that the weekly demand function for this product is p=500-2x. Find the number of units that will give the break even points for the product. Find the number of units that will give maximum profit.
To find the break-even point, we need to determine the quantity of units where the total cost equals the total revenue. This occurs when the profit is zero.
Step 1: Set Profit (π) equal to zero
Profit = Revenue - Cost
In a monopoly market, revenue is calculated as the product of price (p) and quantity (x):
Revenue = p * x
Total Cost is given by the function C(x) = 3600 + 100x + 2x^2.
Step 2: Add the profit equation:
0 = (p * x) - C(x)
Step 3: Substitute the demand function for price:
0 = (500 - 2x) * x - C(x)
Step 4: Substitute the cost function for C(x):
0 = (500 - 2x) * x - (3600 + 100x + 2x^2)
Step 5: Simplify and solve the equation:
0 = 500x - 2x^2 - 3600 - 100x - 2x^2
0 = -4x^2 + 400x - 3600
Step 6: Rearrange the equation to solve for x:
4x^2 - 400x + 3600 = 0
Step 7: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, factoring may be the most straightforward way:
4(x^2 - 100x + 900) = 0
(x - 10)(x - 90) = 0
Step 8: Set each factor equal to zero to find the possible values for x:
x - 10 = 0 --> x = 10
x - 90 = 0 --> x = 90
Therefore, the break-even points for the product are at x = 10 units and x = 90 units.
To find the number of units that will give maximum profit, you need to determine the quantity that maximizes the profit function. The profit function is the difference between total revenue and total cost.
Step 1: Calculate the profit function:
Profit = Revenue - Cost
Profit = (p * x) - C(x)
Step 2: Substitute the demand function and cost function:
Profit = ((500 - 2x) * x) - (3600 + 100x + 2x^2)
Step 3: Simplify the equation:
Profit = 500x - 2x^2 - 3600 - 100x - 2x^2
Profit = -4x^2 + 400x - 3600
Step 4: To find the maximum profit, take the derivative of the profit function with respect to x, and set it to zero:
d(Profit)/dx = 0
Step 5: Calculate the derivative:
d(Profit)/dx = -8x + 400
Step 6: Set the derivative equal to zero and solve for x:
-8x + 400 = 0
-8x = -400
x = 50
The number of units that will give maximum profit is x = 50 units.