Problem 76 - Algebra and Trigonometry

Question

Manufacturing Cost The manager of a furniture factory finds that it costs \(\$ 2200\) to manufacture 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 an equation that expresses this relationship. Then graph the equation. (b) What is the slope of the line in part (a), and what does it represent? (c) What is the \(y\) -intercept of this line, and what does it represent?

Step-by-Step Solution

Answer

(a) The equation is \(y = 13x + 900\). (b) The slope is 13, representing the cost per chair. (c) The y-intercept is 900, representing the fixed manufacturing cost.

Step 1: Define Points

We are given that it costs \( \\(2200 \) to manufacture 100 chairs and \( \\)4800 \) to manufacture 300 chairs. These will be our two points, \((x_1, y_1) = (100, 2200)\) and \((x_2, y_2) = (300, 4800)\) where \(x\) is the number of chairs and \(y\) is the cost.

Step 2: Calculate the Slope

The slope \(m\) of the line can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given values: \[ m = \frac{4800 - 2200}{300 - 100} = \frac{2600}{200} = 13. \] This means for every additional chair, the cost increases by \$13.

Step 3: Find the y-intercept

Using the point-slope form of the equation \(y - y_1 = m(x - x_1)\), we substitute one point, say \((100, 2200)\), and the slope \(m = 13\): \[ y - 2200 = 13(x - 100). \] Simplifying, we find the equation of the line \(y = 13x + 900\). The \(y\)-intercept is \(900\), which represents the fixed cost of manufacturing (the cost when no chairs are produced).

Step 4: Formulate the Linear Equation

From the calculations above, the linear equation representing the cost \(y\) as a function of the number of chairs \(x\) is \(y = 13x + 900\).

Step 5: Graph the Equation (Description)

To graph this equation, plot the two points \((100, 2200)\) and \((300, 4800)\) on a coordinate plane. Draw a straight line through these points, which will extend beyond them to show the linear relationship.

Step 6: Interpret the Slope

The slope of the line, \(m = 13\), represents the variable cost per chair, which means that the cost of producing each additional chair is \$13.

Step 7: Interpret the y-intercept

The \(y\)-intercept of the line is \(900\). This value represents the fixed cost of the company, which are the costs incurred without producing any chairs.

Key Concepts

Slope Calculation

The slope of a line in a linear equation essentially tells us how steep the line is and the direction it goes. When talking about costs, as in our problem with chair manufacturing, the slope helps us understand how much the cost changes with each additional chair produced.
To find this slope, we use two given points that represent the number of chairs produced and the associated cost. The formula to calculate the slope \( m \) is given by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Substituting in our points, \((x_1, y_1) = (100, 2200)\) and \((x_2, y_2) = (300, 4800)\), we get:

• \( m = \frac{4800 - 2200}{300 - 100} = \frac{2600}{200} = 13 \)

This tells us that for every additional chair manufactured, the cost increases by $13.00.
Knowing the slope is crucial because it helps businesses predict and plan for additional costs as production scales up.

Y-Intercept

The y-intercept is a key component in any linear equation such as the one describing our manufacturing costs. It represents the point where the line crosses the y-axis. In real-world terms, it answers the question, "What would the cost be if no chairs are produced?"
To determine the y-intercept, consider a situation where the number of chairs, \( x \), is zero. The y-intercept, \( b \), can be found using the equation of the line \( y = mx + b \). After finding the slope, we can substitute it and one of our points into the equation and simplify to find \( b \).
For our chair manufacturing situation, substituting into point-slope equation with point \((100, 2200)\) and slope \( m = 13 \), we solve:

• \( y - 2200 = 13(x - 100) \)
• Simplifying gives \( y = 13x + 900 \)
• So, the y-intercept \( b = 900 \)

This intercept shows that even with zero production, there are fixed costs of $900, which might include rent, management salaries, or utilities.

Manufacturing Cost

Manufacturing cost is one of the primary considerations for any production company. It is essential to break down these costs to understand both fixed and variable components. From the equation derived, \( y = 13x + 900 \), one can observe both elements clearly.
- **Variable Costs**: These are the costs that vary with production output. Here, the variable cost per chair is represented by the slope, \( m = 13 \). So, for each chair produced, an additional $13 is spent.- **Fixed Costs**: The y-intercept, \( b = 900 \), represents fixed costs. These are expenses that remain constant regardless of the number of chairs produced, such as utility bills, salaries, and rental costs.
Understanding these costs allows for better budgeting and financial forecasting, helping the company to adjust for changes in production levels efficiently.

Linear Relationship

A linear relationship provides a straightforward way to model how two quantities change together. In our manufacturing cost problem, the linear relationship assumes that any increase in chairs produced results in a consistent increase in costs.
By representing this scenario through a linear equation such as \( y = 13x + 900 \), we establish a predictable pattern:

• The cost increases linearly with production.
• Gives a constant and steady slope, meaning each additional chair costs $13 to produce.
• The fixed cost is represented clearly with the y-intercept, offering transparency in expenses.

Such a relationship is particularly beneficial for businesses. It simplifies decision-making processes by allowing predictions on costs associated with changes in production output. Thus, it forms a foundation for strategic planning and analysis in manufacturing settings.