Problem 40 - Intermediate Algebra

Question

Use quadratic functions. Suppose that the equation \(p(x)=-2 x^{2}+280 x-1000\), where \(x\) represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit?

Step-by-Step Solution

Answer

Sell 70 items to maximize profit.

Step 1: Identify the type of equation

The given equation is a quadratic function of the form \(p(x) = ax^2 + bx + c\), where \(a = -2\), \(b = 280\), and \(c = -1000\). This function is a downward-opening parabola since \(a < 0\).

Step 2: Determine the x-value for maximum profit

For a quadratic function in the form \(ax^2 + bx + c\), the x-value that maximizes or minimizes the function is given by \(x = -\frac{b}{2a}\). Substitute \(b = 280\) and \(a = -2\) into the formula: \[x = -\frac{280}{2 \times -2} = \frac{280}{4} = 70.\]

Step 3: Interpret the result

The value \(x = 70\) represents the number of items that should be sold to achieve maximum profit. Since \(p(x)\) describes the profit, maximizing \(p(x)\) corresponds to maximizing the profit made by the business.

Key Concepts

Profit Maximization

Maximizing profit is a key goal for businesses to ensure sustainability and growth. In a business scenario, profit is typically represented as revenues minus costs.
When dealing with quadratic functions, profit maximization involves finding the optimal number of items to sell in order to generate the highest profit. This is where the concept of **vertex of a parabola** is crucial.

To maximize profit, we need to find the highest point on the graph of the quadratic function, which is the vertex in this case.

• If the parabola opens upwards, it has a minimum point. If it opens downwards, it has a maximum point.
• Because the leading coefficient \(a < 0\), the parabola opens downward, indicating there is a maximum profit to be found.

To find this maximum point, apply the formula \(x = -\frac{b}{2a}\). Plugging in the values will give you the optimal number of items to sell for profit maximization. Remember, it's one thing to calculate this, but practical application involves also considering costs, potential demand, and market trends.

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in various real-world applications, such as physics, engineering, and economics.
A quadratic equation takes the standard form \(ax^2 + bx + c = 0\). Here, \(a\) is the coefficient of the quadratic term, \(b\) is the coefficient of the linear term, and \(c\) is the constant.

The general properties include:

• The highest power of \(x\) is 2, indicating it is a second-degree polynomial.
• The graph of a quadratic equation is a parabola.

Finding the solutions or roots of a quadratic equation can be done using several methods:

• Factoring
• Completing the square
• Quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\))

These solutions give insights into where the parabola intersects the x-axis. However, not all quadratics will have real roots; sometimes these can be complex numbers when the discriminant (^2 - 4ac) is negative.

Parabolas

Parabolas are U-shaped curves that can open either upwards or downwards, depending on the sign of the leading coefficient in their quadratic equation.
They are symmetrical around a vertical axis, called the axis of symmetry, which passes through the vertex of the parabola.

• When the coefficient \(a > 0\), the parabola opens upwards.
• Conversely, when \(a < 0\), it opens downwards, like in our profit maximization example.

Parabolas are described by the equation \(y = ax^2 + bx + c\), and have several key features:

• Vertex: The highest or lowest point on the parabola. Calculated using the formula \(x = -\frac{b}{2a}\).
• Axis of Symmetry: An imaginary line that divides the parabola into two symmetrical halves.
• Focus and Directrix: Geometric properties essential for understanding reflections and other applications, though not usually needed for simple quadratic problems.

Understanding parabolas is crucial, especially in disciplines that involve trajectory calculations and optimizations.