Questions: Suppose that c(x)=4x^3-30x^2+5,029x is the cost of manufacturing x items. If possible, find a production level that will minimize the average cost of making x items. A. There is not a production level that will minimize average cost. B. 5,029 items C. 89 items D. 54 items

Suppose that c(x)=4x^3-30x^2+5,029x is the cost of manufacturing x items. If possible, find a production level that will minimize the average cost of making x items.
A. There is not a production level that will minimize average cost.
B. 5,029 items
C. 89 items
D. 54 items

Transcript text: Suppose that $c(x)=4 x^{3}-30 x^{2}+5,029 x$ is the cost of manufacturing $x$ items. If possible, find a production level that will minimize the average cost of making $x$ items. A. There is not a production level that will minimize average cost. B. 5,029 items C. 89 items D. 54 thems

Solution

Solution Steps

To minimize the average cost, we need to find the production level $ x $ that minimizes the average cost function $ \frac{c(x)}{x} $. This involves taking the derivative of the average cost function, setting it to zero, and solving for $ x $.

Step 1: Define the Cost Function

The cost function is given by

\[ c(x) = 4x^3 - 30x^2 + 5029x. \]

Step 2: Calculate the Average Cost Function

The average cost function $ A(x) $ is defined as

\[ A(x) = \frac{c(x)}{x} = \frac{4x^3 - 30x^2 + 5029x}{x} = 4x^2 - 30x + 5029. \]

Step 3: Find the Derivative of the Average Cost Function

To find the production level that minimizes the average cost, we take the derivative of $ A(x) $:

\[ A'(x) = 12x - 30. \]

Step 4: Set the Derivative to Zero

Setting the derivative equal to zero to find critical points:

\[ 12x - 30 = 0 \implies x = \frac{30}{12} = \frac{5}{2} = 15/4. \]