Problem 206 - Elementary Algebra

Question

In the following exercises, solve. Round answers to the nearest tenth. A retailer who sells backpacks estimates that, by selling them for \(x\) dollars each, he will be able to sell \(100-x\) backpacks a month. The \(\quad\) quadratic \(\quad\) equation \(R=-x^{2}+100 x\) is used to find the \(R\) received when the selling price of a backpack 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 revenue is 50 dollars, and the maximum revenue is 2500 dollars.

Step 1: Identify the quadratic equation

The given quadratic equation is \( R = -x^{2} + 100x \), where \(R\) represents the revenue, and \(x\) is the selling price of each backpack.

Step 2: Identify the coefficients

In the equation \( R = -x^{2} + 100x \), identify the coefficients: \(a = -1\), \(b = 100\).

Step 3: Find the vertex of the parabola

The maximum revenue occurs at the vertex of the parabola. The formula for the vertex (\(x\)-coordinate) of the quadratic function \(ax^{2} + bx + c\) is \( x = -\frac{b}{2a} \). Substituting the coefficients, we get \( x = -\frac{100}{2(-1)} = 50 \). Therefore, the selling price that maximizes the revenue is \(x = 50\) dollars.

Step 4: Calculate the maximum revenue

Substitute \(x = 50\) into the original equation to find the maximum revenue. \( R = - (50)^{2} + 100 (50) = -2500 + 5000 = 2500 \). So, the maximum revenue is \(2500\) dollars.

Step 5: Round the answer

Since the answer is already an integer (2500), rounding to the nearest tenth is not necessary.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c \), where:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

A quadratic equation graphs as a parabola.
The standard form is helpful in identifying the key components used in solving problems.
In our exercise:

• The quadratic equation is \( R = -x^2 + 100x \)
• \( R \) represents revenue
• \( x \) represents the selling price per backpack

Understanding the structure of quadratic equations allows us to predict the shape and properties of their graphs.

Vertex Formula

When solving problems involving quadratic equations, finding the vertex is essential.
The vertex formula helps us find the turning point of the parabola, where it reaches either a maximum or minimum.

The Vertex Formula:

To find the \( x \)-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]In our case:
\( R = -x^2 + 100x \)

• \( a = -1 \)
• \( b = 100 \)

Substituting in:
\[ x = -\frac{100}{2(-1)} = 50 \]The vertex helps us determine the optimal price point (\( x = 50 \) dollars) for the maximum revenue.

Maximum Revenue Calculation

Calculating the maximum revenue involves substituting the \( x \)-coordinate of the vertex back into the original quadratic equation.
In our problem, we found that the optimal price point is \( x = 50 \).
We substitute \( x = 50 \) into \( R = -x^2 + 100x \):
\[ R = -50^2 + 100(50) \]Simplifying:
\[ R = -2500 + 5000 \]Thus, the maximum revenue is 2500 dollars.

• Price per backpack: 50 dollars
• Maximum Revenue: 2500 dollars

Coefficient Identification

Identifying coefficients in a quadratic equation is a crucial step.
In a standard quadratic equation \( ax^2 + bx + c \), the coefficients are the values in front of the variables.
For our equation \( R = -x^2 + 100x \):

• The coefficient of \( x^2 \) is \( a = -1 \)
• The coefficient of \( x \) is \( b = 100 \)
• The constant term is \( c = 0 \) (not visible in the equation but is important to note)

These coefficients are essential for applying the vertex formula and solving the equation.
Understanding these components will help in correctly applying mathematical formulas and achieving accurate results.