Problem 66 - Algebra for College Students

Question

Solve each problem. A company uses the function \(C(x)=\) \(0.02 x^{2}-3.4 x+150\) to model the unit cost in dollars for producing \(x\) stabilizer bars. For what number of bars is the unit cost at its minimum? What is the unit cost at that level of production?

Step-by-Step Solution

Answer

The unit cost is minimized at 85 bars, where the cost is 5.5 dollars.

Step 1: Identify the quadratic function

The given cost function is a quadratic equation: \[C(x) = 0.02x^2 - 3.4x + 150\].

Step 2: Determine the vertex form

For a quadratic function in the form \[ax^2 + bx + c\], the vertex, which gives the minimum or maximum point, occurs at \[x = \frac{-b}{2a}\]. Here, \(a = 0.02\) and \(b = -3.4\).

Step 3: Calculate the x-coordinate of the vertex

Substitute the values of \(a\) and \(b\) into the formula to find \(x\): \[x = \frac{-(-3.4)}{2(0.02)} = \frac{3.4}{0.04} = 85\]. Therefore, the unit cost is minimized when producing 85 stabilizer bars.

Step 4: Find the minimum unit cost

Substitute \(x = 85\) back into the original cost function to find the minimum unit cost: \[C(85) = 0.02(85)^2 - 3.4(85) + 150\]. Calculate this: \[C(85) = 0.02(7225) - 3.4(85) + 150 = 144.5 - 289 + 150 = 5.5\]. Thus, the minimum unit cost is 5.5 dollars.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It takes the form \[ ax^2 + bx + c = 0 \]where

• a is the coefficient of the squared term,
• b is the coefficient of the linear term,
• c is the constant term.

The graph of a quadratic equation forms a parabola, which can open either upwards or downwards depending on the coefficient a.

If a is positive, the parabola opens upwards, producing a minimum value. Conversely, if a is negative, the parabola opens downwards, producing a maximum value.

In our exercise, \[C(x) = 0.02x^2 - 3.4x + 150\], this is a quadratic equation where:

• a = 0.02,
• b = -3.4,
• c = 150.

Understanding these components is crucial for solving and analyzing quadratic equations.

Vertex Formula

The vertex of a parabola represented by the quadratic equation \[ ax^2 + bx + c \]is the point where the parabola changes direction. It is the maximum or minimum point of the graph.

To find the x-coordinate of the vertex, we use the vertex formula:\[ x = -\frac{b}{2a} \]Here,

• b and a are coefficients from the quadratic equation.

In our example:\[ a = 0.02 \,\text{and}\, b = -3.4\]Substituting these values into the formula, we get:\[ x = -\frac{-3.4}{2(0.02)} = \frac{3.4}{0.04} = 85 \]Thus, the vertex occurs at x = 85.

The x-coordinate of the vertex helps us determine the number of bars for which the unit cost is at its minimum.

Cost Function

A cost function in mathematics represents the cost of producing a certain number of units of a product. It is often used in economics and business to analyze production costs.

In our exercise, we are given the cost function:\[C(x) = 0.02x^2 - 3.4x + 150 \]Here,

• C(x) denotes the unit cost of producing x stabilizer bars.
• The function is quadratic, indicating that as production increases, costs will initially drop to a minimum and then start to rise again.

To find the minimum cost, we substitute the vertex's x-coordinate back into the original quadratic equation.

Since we found the x-coordinate to be 85 from the previous section, we now calculate the minimum cost:\[ C(85) = 0.02(85)^2 - 3.4(85) + 150 \]
Calculate step-by-step:

• First, \(85^2 = 7225 \),
• next, \(0.02 \times 7225 = 144.5\),
• then, \(3.4 \times 85 = 289\),
• finally, \(144.5 - 289 + 150 = 5.5\).

Thus, the minimum unit cost is \(5.5\) dollars at a production level of \(85\) stabilizer bars.