Problem 72 - Algebra and Trigonometry

Question

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) Equation: \( C = 13x + 900 \); (b) Slope: 13, cost per chair; (c) Y-intercept: 900, fixed cost.

Step 1: Express the linear relationship

To express the linear relationship between cost \( C \) and the number of chairs \( x \), we assume the relationship is of the form \( C = mx + b \). We'll use the given points: \((100, 2200)\) and \((300, 4800)\) to find \( m \) and \( b \).

Step 2: Find the slope (m)

The slope \( m \) is calculated by the formula \( m = \frac{C_2 - C_1}{x_2 - x_1} \). Substituting the given values, \( m = \frac{4800 - 2200}{300 - 100} = \frac{2600}{200} = 13 \). This is the rate of change of cost with each additional chair.

Step 3: Use a point to find the y-intercept (b)

Substitute one of the points and the slope into the equation to find \( b \). Using the point \((100, 2200)\): \[ 2200 = 13\cdot100 + b \]\[ 2200 = 1300 + b \]So, \( b = 900 \).

Step 4: Form the final equation

With \( m = 13 \) and \( b = 900 \), the equation for the relationship is \( C = 13x + 900 \).

Step 5: Graph the equation

Plot the linear equation \( C = 13x + 900 \) on a coordinate plane with \( x \) (number of chairs) on the horizontal axis and \( C \) (cost) on the vertical axis. The line should pass through points \((100, 2200)\) and \((300, 4800)\).

Step 6: Interpret the slope

The slope of 13 represents the additional cost in dollars to produce each additional chair.

Step 7: Interpret the y-intercept

The \( y \)-intercept of 900 represents the fixed cost when no chairs are being produced.

Key Concepts

Slope

In the context of linear equations, the slope is a measure of how steep a line is. It indicates how much one variable changes when the other variable changes by a unit amount. The slope is represented by the letter \( m \) in the equation \( y = mx + b \).

In the exercise, the manager uses two points: \((100, 2200)\) and \((300, 4800)\) to calculate the slope, which comes out to be 13. This value is derived using the formula for slope:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the values we get:\[m = \frac{4800 - 2200}{300 - 100} = \frac{2600}{200} = 13\]This slope of 13 represents the additional cost incurred to produce each extra chair. Practically, every new chair adds \(13 to the production cost.

• The slope provides insight into the variable costs associated with production.
• A greater slope means a higher cost per additional unit.

Understanding the slope is crucial for resource planning and budgeting. Knowing that each extra chair costs \)13 helps the manager predict future costs based on production increases. This way, the business can make informed decisions about scaling up production.

Y-Intercept

The \( y \)-intercept in a linear equation is the value of \( y \) when \( x \) is zero. It gives you the starting point of the line on the vertical axis. In the equation \( y = mx + b \), \( b \) is the y-intercept.

For the factory problem, when no chairs (\( x \)) are produced, the \( y \)-intercept \( b \) is 900. To find this, we substitute one of the points along with the slope into the equation:\[2200 = 13 \cdot 100 + b \]Solving for \( b \) yields:\[2200 = 1300 + b \]\[b = 900\]This value of 900 represents the fixed operating costs the company incurs just by being open, even if no chairs are produced.

• The \( y \)-intercept is crucial for breaking down total costs into fixed and variable components.
• Recognizing fixed costs can assist in determining profitability and pricing strategies.

Understanding the \( y \)-intercept aids in financial forecasting and helps in identifying baseline expenses.

Linear Relationship

A linear relationship between two variables suggests they change at a constant rate to each other. This is typically represented by a straight line in a graph. In algebraic terms, a linear relationship takes the form of \( y = mx + b \).

In the furniture factory problem, the cost of producing chairs was assumed to be a linear relationship with the number of chairs made. The equation \( C = 13x + 900 \) captures this relationship:- \( C \) is the total cost.- \( x \) is the number of chairs produced.- \( 13 \) is the slope—indicating the variable cost per chair.- \( 900 \) is the \( y \)-intercept—representing fixed costs.

• A linear relationship provides a straightforward prediction model: if you know one variable, you can predict the other.
• These relationships are easy to graph and interpret, making them useful for quick decision-making.

In practical terms, understanding this linear relationship assists the manager in planning and optimization efforts, such as how changing production affects costs. It serves as a foundational tool for budgeting and strategic planning.