Problem 44 - Calculus

Question

Residents of the town of Maple Grove who are connected to the municipal water supply are billed a fixed amount monthly plus a charge for each cubic foot of water used. A household using 1000 cubic feet was billed \(\$ 40,\) while one using 1600 cubic feet was billed \(\$ 55\) (a) What is the charge per cubic foot? (b) Write an equation for the total cost of a resident's water as a function of cubic feet of water used. (c) How many cubic feet of water used would lead to a bill of \(\$ 100 ?\)

Step-by-Step Solution

Answer

(a) $0.025 per cubic foot, (b) \( T = 15 + 0.025x \), (c) 3400 cubic feet.

Step 1: Set up the Equations

We need to establish two equations based on the information given. Let's denote the fixed monthly charge as \( F \) and the charge per cubic foot as \( C \). We know that: \[ F + 1000C = 40 \] \[ F + 1600C = 55 \] These are two linear equations based on the two different water usages.

Step 2: Solve for F and C

First, subtract the first equation from the second equation to eliminate \( F \): \[ (F + 1600C) - (F + 1000C) = 55 - 40 \] Simplifying, we get: \[ 600C = 15 \] \[ C = \frac{15}{600} = 0.025 \] Therefore, the charge per cubic foot is \( 0.025 \) dollars.

Step 3: Find the Fixed Charge F

Now that we know \( C = 0.025 \), we can substitute this back into one of the original equations to find \( F \). Using \( F + 1000C = 40 \): \[ F + 1000(0.025) = 40 \] \[ F + 25 = 40 \] \[ F = 15 \] So, the fixed monthly charge is \( 15 \) dollars.

Step 4: Write the Cost Equation

With \( C = 0.025 \) and \( F = 15 \), the equation for the total cost (\( T \)) as a function of cubic feet used (\( x \)) becomes: \[ T = 15 + 0.025x \] This is the equation that describes the total cost for water usage.

Step 5: Calculate Water Usage for a $100 Bill

To find how many cubic feet of water results in a bill of \( 100 \) dollars, set \( T = 100 \) in the equation: \[ 100 = 15 + 0.025x \] Solving for \( x \): \[ 100 - 15 = 0.025x \] \[ 85 = 0.025x \] \[ x = \frac{85}{0.025} \] \[ x = 3400 \] Thus, 3400 cubic feet of water would result in a bill of \( 100 \) dollars.

Key Concepts

fixed charge

In the context of the Maple Grove water billing problem, the fixed charge represents a consistent flat fee that each household pays every month, regardless of their water usage. This amount is termed as "fixed" because it remains the same across bills, no matter how many cubic feet of water are consumed.

Understanding fixed charges is crucial for many utility bills and services. They can be thought of as a baseline cost for being connected to the water system. For instance:

• In our problem, the fixed monthly charge is calculated to be \( 15 \) dollars.
• This charge covers administrative and other basic costs involved in maintaining the water service to households.

Recognizing the concept of a fixed charge helps you better understand billing methodologies, providing insights into how standard utility services are priced.

charge per unit

The charge per unit in this exercise refers to the cost per cubic foot of water used. This is an important variable because it determines how the bill will vary with consumption. Unlike the fixed charge, the charge per unit increases with the amount of water a household uses.

In the Maple Grove exercise, this is a linear cost directly tied to usage:

• The charge per cubic foot is calculated to be \( 0.025 \) dollars, or 2.5 cents.
• This means for every additional cubic foot of water used, the bill increases by this unit rate.

Understanding this concept helps residents manage their water usage more effectively since higher consumption leads to higher bills. Knowing the unit rate allows for better budgeting and conservation efforts.

water usage cost function

The water usage cost function serves as a mathematical model to calculate the total cost of water for any given amount of use. In this scenario, the function integrates both the fixed charge and the charge per unit, making it a linear equation.

The specific function derived from the exercise is:

• \( T = 15 + 0.025x \), where \( T \) is the total cost and \( x \) is the cubic feet of water used.
• This function gives us a clear formula to determine expenses based on usage, combining the fixed monthly fee and the variable charge per unit.

Utilizing such functions enable consumers to forecast their bills and make informed decisions in optimizing their water usage. It's a practical application of linear equations, which is fundamental in financial planning for utilities.