Problem 25 - 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, \(w .\) (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) The formula is \( C = 0.12w + 4.16 \). (b) Slope is \( 0.12 \), cost increases \($0.12\) per gallon. (c) Intercept is \( 4.16 \), a base cost.

Step 1: Identify Given Data Points

We have two points given in the form of (waste in gallons, cost in dollars): \((32, 8)\) and \((68, 12.32)\). These points will help us determine the linear formula.

Step 2: Calculate the Slope

The slope \( m \) of the line can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points \((32, 8)\) and \((68, 12.32)\), we calculate:\[ m = \frac{12.32 - 8}{68 - 32} = \frac{4.32}{36} = 0.12 \]This means the cost increases by \($0.12\) per gallon of waste.

Step 3: Use Point-Slope Form to Find the Line Equation

Using the point-slope form of the line \( y - y_1 = m(x - x_1) \), substitute one of the points, say \( (32, 8) \), and the calculated slope: \( y - 8 = 0.12(x - 32) \). Solve for \( y \) to get: \[ y = 0.12x - 3.84 + 8 \] \[ y = 0.12x + 4.16 \]. Thus, the linear formula is \( C = 0.12w + 4.16 \).

Step 4: Determine the Slope

The slope of the line is \( 0.12 \), which means for each additional gallon of waste, the cost increases by \($0.12\).

Step 5: Determine the Vertical Intercept

The vertical intercept is found when \( w = 0 \) in the equation \( C = 0.12w + 4.16 \). Therefore, the intercept is \( 4.16 \). This represents the fixed starting cost (or base fee) for waste collection, \($4.16\), even if no waste is produced.

Key Concepts

Understanding Slope Calculation

The slope is a key component of linear equations as it represents the rate of change between two variables. In this exercise, we're dealing with the cost of waste collection as it changes with the amount of waste produced.
To find the slope, we use the formula:

• Given points \((x_1, y_1) = (32, 8)\) and \((x_2, y_2) = (68, 12.32)\), to find slope \(m\), you calculate:
• \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12.32 - 8}{68 - 32} = \frac{4.32}{36} = 0.12\]

This slope of 0.12 tells us that for every additional gallon of waste, the cost increases by \\(0.12. Understanding this helps you see how waste production impacts pricing. The units can be seen as \\) per gallon, showing a clear financial relationship.
The slope is vital for making predictions about costs for different levels of waste production.

Exploring the Point-Slope Form

The point-slope form of a line is a handy tool for crafting a linear equation. When you know a point on the line and the slope, you can easily write the equation.
The formula for point-slope form is:

• \(y - y_1 = m(x - x_1)\)
• Insert our slope \(m = 0.12\) and a point on the line \((32, 8)\):
• \[y - 8 = 0.12(x - 32)\]

After simplifying this, we arrive at the linear equation:\[y = 0.12x + 4.16\]
This shows how the cost changes (\(y\)) based directly on the waste produced (\(x\)). The simplicity of point-slope form allows you to quickly transition these values into a functional linear equation that can be used in real-world financial planning scenarios.

Creating a Linear Equation

A linear equation represents a straight line and is usually written in the form \(y = mx + b\) where \(m\) is the slope, and \(b\) is the y-intercept.
From our calculations using point-slope form, we derived:

• \[C = 0.12w + 4.16\]

In this equation:

• \(C\) is the cost of waste collection
• \(w\) is the number of gallons of waste
• The slope (\(0.12\)) indicates how much the cost increases per gallon of waste
• The y-intercept (\(4.16\)) is the base fee when no waste is produced

This linear equation is a practical tool that explicitly shows the relationship between waste generation and collection cost. It highlights both the variable cost \(0.12w\) and the fixed cost \(4.16\), helping you understand and manage the finances with waste collection services effectively.