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Chapter 2: Problem 50

Manufacturing cost The manager of a furniture factory finds that it costs \(\$ 2200\) to produce 100 chairs in one day and \(\$ 4800\) to produce 300 chairs in one day. (a) Assuming that the relationship between cost and the number of chairs produced is linear, find a linear function \(C\) that models the cost of producing \(x\) chairs in one day. (b) Draw a graph of \(C .\) What is the slope of this line? (c) At what rate does the factory's cost increase for every additional chair produced?

Short Answer

Expert verified
(a) \( C(x) = 13x + 900 \). (b) Slope is 13. (c) Cost increases by $13 per chair.

Step by step solution

01

Identify Known Values

We have two points that represent the cost and the number of chairs: (100, 2200) and (300, 4800). Here, the first number in the pair represents the number of chairs, and the second number represents the cost.
02

Calculate Slope of the Line

Using the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]we substitute the known values:\[ m = \frac{4800 - 2200}{300 - 100} = \frac{2600}{200} = 13 \]So, the slope of the line is 13.
03

Determine the Linear Equation

The equation of a line is given by \[ y = mx + b \] where m is the slope. We have m = 13. We use one point, say (100, 2200), to find b:\[ 2200 = 13(100) + b \]\[ 2200 = 1300 + b \]\[ b = 2200 - 1300 = 900 \]Thus, the linear function is \[ C(x) = 13x + 900 \].
04

Graph the Linear Function

To graph the line, plot the two points (100, 2200) and (300, 4800) and draw a straight line through them. The y-intercept is 900, which is the cost when 0 chairs are produced.
05

Interpret the Slope

The slope of the linear function is 13, which represents the rate of change in cost per additional chair produced. This means that for each additional chair, the cost increases by $13.
06

Report Slope and Rate

The slope of the line is 13. Therefore, the factory's cost increases at a rate of $13 for every additional chair produced.

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Key Concepts

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

Slope Calculation
Calculating the slope of a linear function is a key to understanding how changes in one variable affect another. In this exercise, we're dealing with cost changes as the number of chairs produced changes. The slope tells us how quickly the cost increases when the number of chairs increases.
  • We start with finding two points from the cost and production data:
    • (100, 2200)
    • (300, 4800)
  • These points represent the number of chairs and associated costs.
  • The slope formula is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substitute the known values into the formula to find:\[ m = \frac{4800 - 2200}{300 - 100} = 13 \]The slope, 13, indicates that each additional chair increases the production cost by $13.
Graphing Linear Equations
Graphing a linear equation helps visualize the relationship between variables. Here, we graph the cost equation to understand how production influences total cost.
  • The linear equation is \[ C(x) = 13x + 900 \]
  • The y-intercept is 900, representing the base cost when no chairs are produced.
  • We plot significant points:
    • (100, 2200)
    • (300, 4800)
Start by plotting the y-intercept at 900.
Then, use the calculated slope to draw a line through the points to extend the graph.
This visual representation shows costs rising as chairs are produced.
Cost Analysis
Understanding cost analysis with linear functions is essential for financial decisions in manufacturing. This allows managers to predict costs as production scales.
  • The slope from the function \[ C(x) = 13x + 900 \] signifies incremental cost changes.
  • The base cost (y-intercept), 900, occurs with zero production.
  • Each chair adds $13 to the production cost, indicating linear cost growth.
Through this method, managers can forecast how production increases affect total costs.
They can use this data to make informed production and pricing decisions.
This understanding guides strategic planning and budget management.

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