Questions: The total cost of producing a type of tractor is given by C(x)=11000-60 x+0.1 x^2, where x is the number of tractors produced. How many tractors should be produced to incur minimum cost?

The total cost of producing a type of tractor is given by C(x)=11000-60 x+0.1 x^2, where x is the number of tractors produced. How many tractors should be produced to incur minimum cost?

Transcript text: The total cost of producing a type of tractor is given by $C(x)=11000-60 x+0.1 x^{2}$, where $x$ is the number of tractors produced. How many tractors should be produced to incur minimum cost?

Solution

Solution Steps

To find the number of tractors that should be produced to incur the minimum cost, we need to find the vertex of the quadratic function $ C(x) = 11000 - 60x + 0.1x^2 $. The vertex form of a quadratic function $ ax^2 + bx + c $ gives the x-coordinate of the vertex as $ x = -\frac{b}{2a} $. Here, $ a = 0.1 $ and $ b = -60 $.

Solution Approach

Identify the coefficients $ a $ and $ b $ from the quadratic function.
Use the formula $ x = -\frac{b}{2a} $ to find the number of tractors that minimize the cost.

Step 1: Identify the Quadratic Function

The total cost of producing a type of tractor is given by the quadratic function: \[ C(x) = 11000 - 60x + 0.1x^2 \] where $ x $ represents the number of tractors produced.

Step 2: Determine Coefficients

From the quadratic function, we identify the coefficients:

$ a = 0.1 $
$ b = -60 $

Step 3: Calculate the Vertex

To find the number of tractors that minimizes the cost, we use the vertex formula: \[ x = -\frac{b}{2a} \] Substituting the values of $ a $ and $ b $: \[ x = -\frac{-60}{2 \cdot 0.1} = \frac{60}{0.2} = 300 \]