Questions: Solve the problem. Suppose c(x)=x^3-22x^2+30,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost per item of making x items. A) 12 items B) 10 items C) 11 items D) 13 items

Solve the problem.

Suppose c(x)=x^3-22x^2+30,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost per item of making x items.
A) 12 items
B) 10 items
C) 11 items
D) 13 items

Transcript text: Question 51 (1 point) Solve the problem. Suppose $c(x)=x^{3}-22 x^{2}+30,000 x$ is the cost of manufacturing $x$ items. Find a production level that will minimize the average cost per item of making $x$ items. A) 12 items B) 10 items C) 11 items D) 13 items

Solution

Solution Steps

Step 1: Define the Average Cost Function

The cost function is given by $ c(x) = x^{3} - 22x^{2} + 30000x $. The average cost per item is defined as: \[ \text{Average Cost} = \frac{c(x)}{x} = \frac{x^{3} - 22x^{2} + 30000x}{x} = x^{2} - 22x + 30000 \]

Step 2: Find the Derivative of the Average Cost Function

To minimize the average cost, we take the derivative of the average cost function with respect to $ x $: \[ \frac{d}{dx} \left( x^{2} - 22x + 30000 \right) = 2x - 22 \]

Step 3: Solve for Critical Points

Setting the derivative equal to zero to find critical points: \[ 2x - 22 = 0 \implies x = 11 \]

Step 4: Evaluate the Average Cost at the Critical Point

We evaluate the average cost at $ x = 11 $: \[ \text{Average Cost}(11) = 11^{2} - 22 \cdot 11 + 30000 = 29879 \]