Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

A cable television firm presently serves 8000 households and charges $$\$ 50$$ per month. A marketing survey indicates that each decrease of $$\$ 5$$ in the monthly charge will result in 1000 new customers. Let \(R(x)\) denote the total monthly revenue when the monthly charge is \(x\) dollars. (a) Determine the revenue function \(R\). (b) Sketch the graph of \(R\) and find the value of \(x\) that results in maximum monthly revenue.

Step-by-Step Solution

Answer

The revenue function is \( R(x) = -5000x^2 + 10000x + 400000 \). Max revenue occurs at \$45 per month, generating \$405,000.

Step 1: Define the Variables

Let the initial charge per month be \\(50. A decrease of \\)5 corresponds to 1000 additional customers. Let \( x \) represent the number of \$5 decreases. Thus, the monthly charge per customer can be expressed as \( 50 - 5x \). Similarly, the number of households become \( 8000 + 1000x \).

Step 2: Form the Revenue Function

The total monthly revenue \( R(x) \) is given by the product of the number of households and the charge per household. Therefore, \( R(x) = (8000 + 1000x)(50 - 5x) \).

Step 3: Expand the Expression for R(x)

Expand the expression for \( R(x) \) using the distributive property:\[R(x) = (8000)(50) + (8000)(-5x) + (1000x)(50) + (1000x)(-5x)\]Simplifying, we get:\[R(x) = 400000 - 40000x + 50000x - 5000x^2\]Combining like terms gives:\[R(x) = -5000x^2 + 10000x + 400000\]

Step 4: Determine the Vertex of the Parabola

Since \( R(x) \) is a quadratic function that opens downward (as the coefficient of \( x^2 \) is negative), the maximum revenue occurs at the vertex of the parabola. Use the vertex formula \( x = -\frac{b}{2a} \), where \( a = -5000 \) and \( b = 10000 \):\[x = -\frac{10000}{2(-5000)} = 1\]

Step 5: Calculate the Maximum Revenue and Corresponding Charge

The maximum monthly revenue occurs when \( x = 1 \). Substitute back to find the maximum revenue:\[R(1) = -5000(1)^2 + 10000(1) + 400000 = 405000\]When \( x = 1 \), the charge is \( 50 - 5 \times 1 = 45 \) dollars per month.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be represented in the form \( ax^2 + bx + c \), where 'a', 'b', and 'c' are constants, and \( a eq 0 \). A key characteristic of quadratic functions is their U-shaped or inverted U-shaped graphs called parabolas.
Quadratic functions commonly appear in problems involving area, projectile motion, and optimization.

• The highest or lowest point on the graph of a quadratic function is called the vertex.
• The shape of the parabola is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upward, and if 'a' is negative, it opens downward.
• The axis of symmetry is a vertical line that passes through the vertex, controlling the folding line. Its equation is given by \( x = -\frac{b}{2a} \).

Understanding how a quadratic equation is formed and manipulated is crucial for finding solutions to real-world problems, such as maximizing profits or designing structures.

Maximum Revenue

The concept of maximum revenue is vital in businesses as it means generating the highest possible earnings.
In the context of a quadratic revenue function, maximum revenue occurs at the vertex of the parabola.The function derived in this example, \( R(x) = -5000x^2 + 10000x + 400000 \), is a quadratic function representing a revenue model.

• The negative coefficient of \( x^2 \) confirms the parabola opens downward, indicating that the vertex gives the maximum point.
• Maximum revenue isn’t always the same as maximum profit, as it doesn't account for costs and expenses. It's purely how much is being brought in.

To determine maximum revenue, find the value of \( x \) using the formula \( x = -\frac{b}{2a} \). The resulting value is substituted back into the revenue function to find the total revenue at this optimal point. Here, \( R(1) = 405000 \). Knowing the maximum revenue helps businesses decide on pricing strategies that optimize income.

Vertex of Parabola

The vertex of a parabola is a fundamental concept of quadratic functions as it denotes either the peak or trough of the graph.
For our revenue function, finding the vertex helps identify the price point maximizing revenue.The general form \( ax^2 + bx + c \) relies on the vertex formula \( x = -\frac{b}{2a} \). Applying it helps pinpoint the exact 'x' value where the function has its extreme value.

• In our scenario, \( a = -5000 \) and \( b = 10000 \), resulting in \( x = 1 \).
• Knowing the maximum point is vital for analyzing and optimizing various practical situations, such as financial markets or engineering designs.
• The vertex's 'x' coordinate gives the critical point in the context given by the function, like pricing adjustments to maximize income in our example.

Understanding the vertex location aids in making informed decisions that leverage the best outcomes from quadratic behavior.