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Chapter 2: Problem 104

Total revenue, cost, and profit. Using the same set of axes. sketch the graphs of the total-revenue, total-cost, and total. profit functions. $$R(x)=50 x-0.5 x^{2}, \quad C(x)=10 x+3$$

Short Answer

Expert verified
Graphs of R(x), C(x), and P(x) are a downward parabola, a straight line, and another downward parabola, respectively.

Step by step solution

01

Understanding the Functions

Given the total revenue function R(x) = 50x - 0.5x^2 and the total cost function C(x) = 10x + 3.
02

Defining the Total Profit Function

The total profit function P(x) is calculated by subtracting the total cost function from the total revenue function: P(x) = R(x) - C(x).
03

Calculating the Total Profit Function

Substitute the expressions for R(x) and C(x) into P(x): P(x) = (50x - 0.5x^2) - (10x + 3) = 50x - 0.5x^2 - 10x - 3 Combine like terms: P(x) = 40x - 0.5x^2 - 3.
04

Identifying Graph Characteristics

Note that R(x) and P(x) are quadratic functions that open downwards (since the coefficient of x^2 is negative), and C(x) is a straight line.
05

Determining Intercepts and Vertex of Each Function

For R(x), find the x-intercepts by solving 50x - 0.5x^2 = 0 x = 0 or x = 100. The vertex will be at x = 50. For C(x), the y-intercept is at 3 (when x = 0). For P(x), solve 40x - 0.5x^2 - 3 = 0 to find the x-intercepts.
06

Plotting the Functions

On a set of axes, plot R(x) as a downward-opening parabola with intercepts at x = 0 and x = 100 and a vertex at x = 50. Plot C(x) as a straight line starting at y = 3 with a slope of 10. Plot P(x) as a downward-opening parabola based on the calculated profits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Total Revenue
Total Revenue (R) is the total amount of money a company earns from selling its products or services. It is a function of the number of units sold, often denoted by the variable x. In this exercise, the total revenue function is given by:

\( R(x) = 50x - 0.5x^2 \).

Where
  • 50x represents the initial revenue generated per unit.
  • 0.5x^2 represents the quadratic term that accounts for a decrease in revenue as more units are produced and sold.
This function is a quadratic function, meaning its graph will be a parabola. Because the coefficient of \( x^2 \) is negative, the parabola opens downwards.
Total Cost
Total Cost (C) refers to the total expenditure a company incurs to produce and sell a specific number of units. This includes both fixed and variable costs. In the exercise, the total cost function is given by:

\( C(x) = 10x + 3 \).

Where
  • 10x represents the variable cost per unit
  • 3 represents the fixed cost that does not change with the level of production
This function is linear, which means its graph will be a straight line. The slope is 10, showing the marginal cost per unit, and the y-intercept is 3, representing the fixed costs.
Total Profit
Total Profit (P) is the net earnings of a company after subtracting total costs from total revenue. The total profit function can be derived using the expressions for revenue and cost:

\( P(x) = R(x) - C(x) \).

In this case:

\( P(x) = (50x - 0.5x^2) - (10x + 3) = 40x - 0.5x^2 - 3 \).

Like the total revenue, the total profit function is also quadratic and opens downwards for the same reasons (a negative \( x^2 \) coefficient). It shows how both revenue generation and costs interact to determine profitability.
Quadratic Functions
Quadratic functions are a type of polynomial that can be represented as \( ax^2 + bx + c \), where a, b, and c are constants and \( a eq 0 \). The graph of a quadratic function is a parabola, which can open either upwards or downwards.

Key features of quadratic functions include:
  • The vertex, which is the maximum or minimum point
  • The axis of symmetry, a vertical line passing through the vertex
  • The direction of opening determined by the sign of a: positive means upwards, negative means downwards
In the exercise, the total revenue (\( R(x) \)) and total profit (\( P(x) \)) functions are quadratic and both open downwards.
Graphing Functions
Graphing is the process of plotting functions on a coordinate system to visualize their behaviors. Here's a brief guide on how to graph the functions from the exercise:

For \( R(x) = 50x - 0.5x^2 \):
  • Find the x-intercepts by solving \( R(x) = 0 \)
  • Determine the vertex using the formula \( x = -b/(2a) \)
  • Sketch the downward-opening parabola
For \( C(x) = 10x + 3 \):
  • Identify the y-intercept (constant term)
  • Graph the straight line using the slope (10)
For \( P(x) = 40x - 0.5x^2 - 3 \):
  • Find the x-intercepts and vertex
  • Graph this downward-opening parabola
Combining these graphs on one set of axes illustrates the relationships between revenue, cost, and profit clearly.

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