Question

A liquid form of penicillin manufactured by a pharmaceutical company is sold in bulk at a price of \(\$200\) per unit. If the total production cost (in dollars) for \(x\) units is \[ C(x)=500{,}000+80x+0.003x^2, \] and the production capacity of the firm is at most \(30{,}000\) units in a specified time, then how many units of penicillin must be manufactured and sold during that specified time to maximize the profit? 

• A. 10,000
• B. 20,000
• C. 30,000
• D. 25,000

Hint:

Profit is maximized when Marginal Revenue (\(MR\)) equals Marginal Cost (\(MC\)):
- \(MR = \frac{d}{dx}(200x) = 200\)
- \(MC = \frac{d}{dx}(C(x)) = 80 + 0.006x\)
Set \(MR = MC \implies 200 = 80 + 0.006x \implies 120 = 0.006x \implies x = 20,000\).
This is a standard and fast economic approach.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To maximize profit, we must construct the profit function, which is the difference between total revenue and total cost.
We then use differential calculus to find the production level \(x\) that maximizes this function.

Step 2: Key Formula or Approach:
1. Total Revenue, \(R(x) = \text{Price} \times x\).
2. Profit, \(\Pi(x) = R(x) - C(x)\).
3. To find the maximum, set the first derivative of profit to zero (\(\Pi'(x) = 0\)) and ensure the second derivative is negative (\(\Pi''(x) < 0\)).

Step 3: Detailed Explanation:
We are given:
- Selling price per unit, \(p = \$200\)
- Total Cost, \(C(x) = 500,000 + 80x + 0.003x^2\)
- Maximum capacity constraint: \(x \le 30,000\)
First, construct the Total Revenue function:
\[ R(x) = 200x \]
Next, write down the Profit function \(\Pi(x)\):
\[ \Pi(x) = R(x) - C(x) \]
\[ \Pi(x) = 200x - (500,000 + 80x + 0.003x^2) \]
\[ \Pi(x) = -0.003x^2 + 120x - 500,000 \]
To find the critical point, differentiate \(\Pi(x)\) with respect to \(x\) and set the derivative to zero:
\[ \Pi'(x) = \frac{d}{dx}\left(-0.003x^2 + 120x - 500,000\right) = 0 \]
\[ -0.006x + 120 = 0 \]
Solve this equation for \(x\):
\[ 0.006x = 120 \]
\[ x = \frac{120}{0.006} = \frac{120,000}{6} = 20,000 \]
Now, let us perform the second derivative test to confirm this point is a maximum:
\[ \Pi''(x) = \frac{d^2 \Pi}{dx^2} = -0.006 \]
Since the second derivative is strictly negative (\(\Pi''(x) < 0\)), the profit function reaches a local maximum at \(x = 20,000\).
This value is within the firm's production capacity constraint of 30,000 units.
Thus, the firm must produce and sell 20,000 units to maximize profit.

Step 4: Final Answer:
Therefore, the correct option is (B).