A drug manufactured by a pharmaceutical company is sold in bulk at a price of $150
per unit. The total production cost (in dollars) for x units in one week is C(x) =
0.02x^2 + 100x + 3000. How many units of the drug must be manufactured and sold in
a week to maximize the profit? What is the maximum profit?
Since profit is revenue less cost, we have
p(x) = 150x - (0.02x^2 + 100x + 3000)
now just find the vertex of that parabola.
To find the number of units that must be manufactured and sold in a week to maximize profit, we need to determine the profit function and then find the maximum point.
The profit function can be calculated by subtracting the production cost per unit from the selling price per unit. The selling price per unit is $150, and the production cost function is given by C(x) = 0.02x^2 + 100x + 3000.
So, the profit function, P(x), is:
P(x) = Revenue - Cost
P(x) = (Selling Price per unit * x) - C(x)
P(x) = (150 * x) - (0.02x^2 + 100x + 3000)
To find the maximum profit, we need to find the vertex of the profit function. The vertex of a quadratic function in the form ax^2 + bx + c can be found using the formula x = -b/2a. In our case, a = -0.02, b = 150, and c = (-100x - 3000).
Using the formula x = -b/2a, we get:
x = -150 / (2 * -0.02)
x = 3750
So, 3750 units of the drug must be manufactured and sold in a week to maximize profit.
To find the maximum profit, we substitute the value of x into the profit function:
P(3750) = (150 * 3750) - (0.02(3750^2) + 100 * 3750 + 3000)
P(3750) = 562500 - 281250 + 375000 + 3000
P(3750) = 656250
Therefore, the maximum profit is $656,250.