Problem 41 - 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) \(0.12\) dollars per gallon. (c) \(4.16\) dollars (base cost).

Step 1: Understanding the Linear Relationship

We are given two points on the cost vs. gallons line: \((w_1, C_1) = (32, 8)\) and \((w_2, C_2) = (68, 12.32)\). The cost, \(C\), changes with the number of gallons, \(w\). Our task is to find the equation of this line, which can be written in the form \(C = mw + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Calculate the Slope (m)

The slope of the line, \(m\), can be calculated using the formula: \[ m = \frac{C_2 - C_1}{w_2 - w_1} \]Plugging in the values, we have:\[ m = \frac{12.32 - 8}{68 - 32} = \frac{4.32}{36} = 0.12 \] This means the slope \(m\) is 0.12, and represents the rate of cost increase per additional gallon of waste.

Step 3: Calculate the Y-Intercept (b)

To find the y-intercept \(b\), we use one of the points, \((32, 8)\), along with the slope calculated in Step 2. Set up the equation for \(C\): \[ 8 = 0.12 \times 32 + b \] Solving for \(b\), we get: \[ 8 = 3.84 + b \] \[ b = 8 - 3.84 = 4.16 \]Thus, the y-intercept \(b\) is 4.16.

Step 4: Formulate the Linear Equation

Using the slope \(m = 0.12\) and y-intercept \(b = 4.16\), the linear formula for the cost, \(C\), as a function of gallons, \(w\), is:\[ C = 0.12w + 4.16 \]

Step 5: Interpret the Slope

The slope of the equation, 0.12, means that for each additional gallon of waste, the cost increases by \(0.12\) dollars. This is the incremental cost per gallon in terms of the waste collection service.

Step 6: Interpret the Y-Intercept

The y-intercept, 4.16, indicates the base cost of the waste collection service when no waste is collected. It represents fixed costs not dependent on the amount of waste.

Key Concepts

Slope Interpretation

The slope of a linear equation is a crucial component to understand when analyzing cost functions. In the context of a cost function for waste collection, the slope tells us how the cost changes with each additional unit of waste. Here, the slope is calculated by taking the difference in cost between two points and dividing it by the difference in the number of gallons of waste handled. Specifically, the formula for slope is given by: \[m = \frac{C_2 - C_1}{w_2 - w_1}\]In our example, when we apply the data points (32 gallons, \(8) and (68 gallons, \)12.32), the slope, \(m\), is found to be 0.12.

• This means for each additional gallon of waste, the cost increases by $0.12.
• The units of the slope are dollars per gallon.

This interpretation of the slope is vital as it allows residents and service providers to predict their costs based on the amount of waste generated. Additionally, understanding the slope helps in analyzing how efficiently services handle additional waste without significantly increasing costs.

Y-Intercept Interpretation

In a linear equation representing a cost function, the y-intercept provides significant insights. It represents the initial cost of the service when no units of the product—in this case, waste—are used. For our waste collection cost function, the y-intercept \(b\) can be found using the following equation:\[C = mw + b\]Using the point (32 gallons, $8) and rearranging, we calculate:\[8 = 0.12 \times 32 + b\]Solving for \(b\) gives a y-intercept of 4.16.

• This value represents the fixed cost associated with the waste collection service.
• It is the baseline amount charged regardless of waste production.

The y-intercept is crucial, especially in cost analysis, as it tells us about the non-variable, or fixed costs. These are the expenses the service incurs even when no waste is collected, such as administrative and basic operational costs.

Cost Function Analysis

When dealing with cost functions, we create a formula to help understand and predict costs. In our example of waste collection, the cost function is expressed as a linear equation: \[C = 0.12w + 4.16\]where \(C\) represents the total cost and \(w\) represents the gallons of waste. This linear equation consists of two parts:

• The slope \(0.12\), which represents the increase in cost per additional gallon of waste.
• The y-intercept \(4.16\), representing the fixed base cost.

Cost function analysis allows users, such as households or businesses, to plan and budget for expected costs by understanding how costs will scale with usage.

• It helps in making informed decisions by showing how costs scale with waste.
• Having a clear formula makes it easier to adjust waste strategies to minimize unnecessary costs.

Understanding and interpreting the cost function not only aids in current budgeting but also assists in forecasting future expenses with changes in waste habits.