Problem 24 - Applied Calculus

Question

In a California town, the monthly charge for waste collection is \$ 8\( for 32 gallons of waste and \$ 12.32\) for 68 gallons of waste. (a) Find a linear formula for the cost, \(C\), of waste collection as a function of the number of gallons of waste, \boldsymbol{w}$ (b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection. (c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection.

Step-by-Step Solution

Answer

(a) \( C = 0.12w + 4.16 \); (b) The slope is 0.12 dollars/gallon, representing additional cost per gallon; (c) The intercept is \$4.16, the base charge.

Step 1: Set Up the Linear Equation

We know the formula for a line is given by \( C = mw + b \), where \( C \) is the cost, \( m \) is the slope, \( w \) is the number of gallons of waste, and \( b \) is the y-intercept. We have two points: \((32, 8)\) and \((68, 12.32)\).

Step 2: Calculate the Slope

The slope of the line, \( m \), is calculated using the formula: \( m = \frac{C_2 - C_1}{w_2 - w_1} \). Substituting the provided values: \( m = \frac{12.32 - 8}{68 - 32} = \frac{4.32}{36} = 0.12 \) dollars per gallon. This means for every additional gallon of waste, it costs an extra \( \$0.12 \).

Step 3: Find the Y-Intercept

To find the y-intercept \( b \), we can substitute one of the points back into the line equation. Using point \((32, 8)\), we get: \( 8 = 0.12 \times 32 + b \). Solving for \( b \) gives: \( 8 = 3.84 + b \); hence, \( b = 8 - 3.84 = 4.16 \). So, \( b = 4.16 \), implying the base charge is \$4.16, regardless of waste.

Step 4: Write the Linear Function

Substitute the slope and intercept into the equation of the line to express \( C \) in terms of \( w \): \( C = 0.12w + 4.16 \).

Step 5: Interpret the Slope

The slope \( 0.12 \) with units of dollars per gallon represents the increase in the waste collection cost for each additional gallon of waste.

Step 6: Interpret the Y-Intercept

The y-intercept \( 4.16 \) with units of dollars represents the fixed minimum charge for waste collection even if no waste is collected.

Key Concepts

Slope Calculation

The slope in a linear equation is a crucial element. It reflects the rate at which the dependent variable changes with respect to the independent variable. In the context of the waste collection problem, the slope represents the change in cost as the quantity of waste changes.

To calculate the slope, we use the formula:

• \( m = \frac{C_2 - C_1}{w_2 - w_1} \)

Given the points \( (32, 8) \) and \( (68, 12.32) \), the slope \( m \) is calculated as:

• \( m = \frac{12.32 - 8}{68 - 32} = \frac{4.32}{36} \)
• This simplifies to \( m = 0.12 \)

The units for the slope are dollars per gallon, implying that for each additional gallon of waste, the cost increases by \( $0.12 \). This simple ratio gives a clear understanding of how costs scale with waste volume.

Y-Intercept

The y-intercept in a linear equation is the initial value or starting point of the function when the independent variable is zero.

In our waste collection scenario, the y-intercept symbolizes the base fee charged, regardless of the amount of waste. To determine the y-intercept \( b \), we use the linear formula:

• Insert one known data point (for example, \( (32, 8) \)) into the equation \( C = mw + b \)
• With \( m = 0.12 \), substitute and solve: \( 8 = 0.12 \times 32 + b \)
• This results in: \( b = 8 - 3.84 = 4.16 \)

Thus, the y-intercept is \( $4.16 \), indicating a minimum charge even if no waste is collected. This fixed cost is essential, as it covers basic services unrelated to the amount of waste collected.

Cost Function Analysis

In analyzing the cost function, we express the relationship between gallons of waste collected and the total collection cost.

The equation obtained, \( C = 0.12w + 4.16 \), reveals two vital components:

• The variable component: \( 0.12w \) which accounts for the cost increase with more waste
• The fixed component: \( 4.16 \), ensuring a base charge regardless of usage

This function gives users clarity on how their waste amount affects overall charges. Analyzing it helps predict costs accurately based on waste projections.
The linear nature helps with straightforward calculations, making this approach efficient for both service providers and customers to understand potential cost changes with different waste levels.