Problem 205 - Elementary Algebra

Question

In the following exercises, solve. Round answers to the nearest tenth. A computer store owner estimates that by charging \(x\) dollars each for a certain computer, he can sell \(40-x\) computers each week. The quadratic \(\quad\) equation \(R=-x^{2}+40 x\) is used to find the revenue, \(R,\) received when the selling price of a computer is \(x\). Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

Step-by-Step Solution

Answer

The selling price that maximizes the revenue is \(20\) dollars, and the maximum revenue is \(400\) dollars.

Step 1: Identify the given quadratic equation

The quadratic equation provided is \[R = -x^2 + 40x\]This equation represents the revenue, \(R\), as a function of the selling price, \(x\).

Step 2: Determine the vertex of the parabola

The vertex form of a quadratic equation \(ax^2 + bx + c\) can be used to find the maximum or minimum value. Since our equation is \(R = -x^2 + 40x\), it has a downward-facing parabola. The vertex provides the maximum revenue value. To find the x-coordinate of the vertex, use:\[x = -\frac{b}{2a}\]For the equation \(R = -x^2 + 40x\), \(a = -1\) and \(b = 40\). Plug these values in:\[x = -\frac{40}{2(-1)} = 20\]Thus, the selling price that gives the maximum revenue is \(20\) dollars.

Step 3: Calculate the maximum revenue

To find the maximum revenue, substitute \(x = 20\) back into the revenue equation:\[R = -x^2 + 40x\]\[R = -20^2 + 40 \cdot 20\]\[R = -400 + 800\]\[R = 400\]Thus, the maximum revenue is \(400\) dollars.

Key Concepts

Maximum Revenue

Understanding maximum revenue in the context of quadratics involves recognizing that revenue can be modeled using a quadratic equation. In this case, the computer store owner's revenue is modeled by the equation R = -x^2 + 40x. This equation is a downward-facing parabola, meaning it reaches a peak point (maximum revenue) before decreasing. The goal here is to find the selling price ( x ) that yields the highest revenue ( R ). The maximum point of a downward-facing parabola is always at its vertex.

Vertex of a Parabola

To find the vertex of a parabola, we need to understand the structure of a quadratic equation in the standard form ax^2 + bx + c . The vertex formula x = -\frac{b}{2a} is crucial here. For the given equation R = -x^2 + 40x , the coefficients are a = -1 and b = 40 . Using the vertex formula:\[ x = -\frac{40}{2(-1)} = 20 \]So, the vertex x-coordinate is 20. This means that the store should charge 20 dollars per computer to achieve the maximum revenue. The vertex gives us the selling price where the revenue is at its peak.

Solving Quadratic Equations

Solving quadratic equations efficiently is essential for various applications, including finding maximum revenue. Here, we need to calculate the exact revenue at the vertex point, x = 20 . Plug this value back into the original revenue equation R = -x^2 + 40x :\[ R = -20^2 + 40 \cdot 20 \]\[ R = -400 + 800 \]\[ R = 400 \]Thus, the maximum revenue is 400 dollars. Breaking the solution process into steps helps ensure each part is clearly understood and correctly applied. The vertex gives the maximum value for the quadratic function in contexts like revenue modeling.