91Ó°ÊÓ

In mathematics, a cost function represents the total cost incurred for manufacturing a certain number of units of a product. It is usually expressed as a function of the quantity of goods produced, denoted by a variable such as \(x\). In this exercise, the cost function is given by \(C(x) = x^2 - 15x + 50\). This is a quadratic equation, where:

• \(x\) is the number of units produced
• \(x^2\) represents the increasing costs as production ramps up
• \(-15x\) accounts for linear changes in cost, perhaps due to efficiency improvements
• \(50\) is a fixed cost, unaffected by the number of units produced

By setting the cost function equal to a certain value, such as \(9500\) dollars in this exercise, we can determine how many units need to be produced to reach that cost.

The quadratic formula is a fundamental tool used to solve quadratic equations, which are polynomial equations of the form \(ax^2 + bx + c = 0\). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides solutions for \(x\) by calculating the roots of the equation. In the context of the exercise, the quadratic equation \(x^2 - 15x - 9450 = 0\) is solved using this formula.
Here:

• \(a = 1\)
• \(b = -15\)
• \(c = -9450\)

By plugging these values into the formula, we can solve for the possible values of \(x\) that satisfy the equation, finding how many units will cost \(9500\) dollars.

The discriminant is part of the quadratic formula, located under the square root: \(b^2 - 4ac\). It is used to determine the nature of the roots of a quadratic equation. The value of the discriminant tells us:

• If it's positive, the quadratic equation has two distinct real solutions.
• If it's zero, there is exactly one real solution (a repeated root).
• If it's negative, there are no real solutions, only complex solutions.

In this exercise, the discriminant is calculated as \(38025\), which is positive. This indicates that the equation \(x^2 - 15x - 9450 = 0\) has two different real solutions. Knowing the sign of the discriminant helps predict how many actual solutions to look for and interpret in practical problems such as this.

Real solutions to a quadratic equation are the values that satisfy the equation when substituted back into it. These solutions can be found using the quadratic formula, provided the discriminant is non-negative.
In this exercise, after calculating the discriminant and confirming it is positive, we proceed with the quadratic formula.Using \(b^2 - 4ac = 38025\), the roots were calculated:

• \(x_1 = 105\)
• \(x_2 = -90\)

Since \(x\) represents the number of units manufactured, only non-negative solutions can be considered valid. Thus, \(x = 105\) is the sensible real solution. This corresponds to producing 105 units at the specified cost of \(9500\) dollars. Understanding which solutions are feasible requires analyzing the context alongside the mathematical result.