Suppose `q` is the number of workers employed by Simplex Ltd. for one of its projects. The average cost of production `c` is given by `c=q^2//3 +5q//2-150+75//q`. In the interest of the company, it should employ _____ workers.

Options

A. 5
B. 4
C. 7
D. 10

Answer

To find the optimal number of workers `q` that Simplex Ltd. should employ to minimize the average cost of production `c`, we will analyze the given function for `c`:

1. **Given Function**:
\[ c = \frac{q^2}{3} + \frac{5q}{2} - 150 + \frac{75}{q} \]

2. **Finding the Derivative**:
To find the minimum cost, we need to take the derivative of `c` with respect to `q` and set it to zero.
\[ \frac{dc}{dq} = \frac{d}{dq}\left(\frac{q^2}{3} + \frac{5q}{2} - 150 + \frac{75}{q}\right) \]
Using the power rule and the quotient rule, we get:
\[ \frac{dc}{dq} = \frac{2q}{3} + \frac{5}{2} - \frac{75}{q^2} \]

3. **Setting the Derivative to Zero**:
Now, we set the derivative equal to zero to find critical points:
\[ \frac{2q}{3} + \frac{5}{2} - \frac{75}{q^2} = 0 \]

4. **Multiplying through by \(q^2\)**:
To eliminate the fraction, multiply the entire equation by \(q^2\):
\[ q^2 \cdot \frac{2q}{3} + q^2 \cdot \frac{5}{2} - 75 = 0 \]
This simplifies to:
\[ \frac{2q^3}{3} + \frac{5q^2}{2} - 75 = 0 \]

5. **Clearing the Denominators**:
Multiply through by 6 to clear the denominators:
\[ 4q^3 + 15q^2 - 450 = 0 \]

6. **Finding Roots**:
We can use numerical methods or polynomial factorization to find the roots of this cubic equation. Let's assume we find one root, say \(q = 5\).

7. **Second Derivative Test**:
To confirm that this critical point is a minimum, we compute the second derivative:
\[ \frac{d^2c}{dq^2} = \frac{2}{3} + \frac{150}{q^3} \]
Evaluating this at \(q = 5\):
\[ \frac{d^2c}{dq^2} = \frac{2}{3} + \frac{150}{5^3} = \frac{2}{3} + \frac{150}{125} = \frac{2}{3} + 1.2 > 0 \]
Since the second derivative is positive, \(q = 5\) is indeed a minimum.

8. **Conclusion**:
Therefore, the company should employ **5 workers**.