A manufacturer produces computers data storage floppies at the rate of x units per week and his total cost of production and marketing is `C (x) = (x^(3)/(3)- 10 x^(2)+ 15 x +30)` . Find the slope of average cost .

Answer

To find the slope of the average cost given the total cost function \( C(x) = \frac{x^3}{3} - 10x^2 + 15x + 30 \), we will follow these steps:

### Step 1: Define the Average Cost
The average cost \( A(x) \) is defined as the total cost \( C(x) \) divided by the number of units produced \( x \):
\[ A(x) = \frac{C(x)}{x} \]

### Step 2: Substitute the Cost Function
Substituting the given cost function into the average cost formula:
\[ A(x) = \frac{\frac{x^3}{3} - 10x^2 + 15x + 30}{x} \]

### Step 3: Simplify the Average Cost
We can simplify \( A(x) \) by dividing each term in the numerator by \( x \):
\[ A(x) = \frac{x^3}{3x} - \frac{10x^2}{x} + \frac{15x}{x} + \frac{30}{x} \]
This simplifies to:
\[ A(x) = \frac{x^2}{3} - 10x + 15 + \frac{30}{x} \]

### Step 4: Find the Derivative of Average Cost
To find the slope of the average cost, we need to differentiate \( A(x) \) with respect to \( x \):
\[ A'(x) = \frac{d}{dx}\left(\frac{x^2}{3}\right) - \frac{d}{dx}(10x) + \frac{d}{dx}(15) + \frac{d}{dx}\left(\frac{30}{x}\right) \]

### Step 5: Calculate Each Derivative
Calculating each term:
- The derivative of \( \frac{x^2}{3} \) is \( \frac{2x}{3} \).
- The derivative of \( 10x \) is \( 10 \).
- The derivative of \( 15 \) is \( 0 \).
- The derivative of \( \frac{30}{x} \) can be rewritten as \( 30x^{-1} \), and its derivative is \( -\frac{30}{x^2} \).

Putting it all together:
\[ A'(x) = \frac{2x}{3} - 10 + 0 - \frac{30}{x^2} \]

### Step 6: Final Expression for the Slope of Average Cost
Thus, the slope of the average cost is:
\[ A'(x) = \frac{2x}{3} - 10 - \frac{30}{x^2} \]

### Summary of the Solution
The slope of the average cost function \( A(x) \) is given by:
\[ A'(x) = \frac{2x}{3} - 10 - \frac{30}{x^2} \]