Problem 24 - Prealgebra

Question

Use the following information. A standard shower head uses about 6 gallons of water per minute. The table shows the relationship between time and the water used. (TABLE CANNOT COPY) Given \(m,\) the number of minutes, write an equation that can be used to find \(g\) the number of gallons used.

Step-by-Step Solution

Answer

The equation is \( g = 6m \).

Step 1: Determine the Unit Rate

Identify the rate at which the standard shower head uses water. It is given that a standard shower head uses 6 gallons of water per minute. This unit rate is crucial as it will serve as the slope in our linear equation.

Step 2: Assign Variables

Define the variables needed to express the relationship. In this problem, let \( m \) represent the number of minutes, and \( g \) represent the number of gallons of water used.

Step 3: Formulate the Equation

Using the unit rate and the defined variables, create an equation that relates \( m \) and \( g \). Since the shower head uses 6 gallons per minute, the relationship can be expressed as \( g = 6m \). This equation states that the total gallons used, \( g \), is equal to 6 times the number of minutes, \( m \).

Key Concepts

Understanding Unit Rate in Linear Equations

In the context of linear equations, a "unit rate" helps us understand how one quantity changes with respect to another, specifically when one unit of measurement is involved. It's a crucial component when forming relationships in real-world situations. Suppose you hear that a standard shower head uses 6 gallons of water per minute. This 6 gallons per minute is the unit rate.

Unit rates simplify many types of calculations. It helps us identify constant rates of change. Here, the unit rate tells us exactly how many gallons are used for every minute the shower runs, which can be a valuable insight if you're attempting to manage or reduce water usage. Knowing the unit rate allows us to determine this consistently, without needing to track each individual minute separately.

To use the unit rate in forming a linear equation, the unit rate acts as a multiplier for the independent variable, often representing time. This is because unit rates essentially describe how much one unit of your first variable affects the second variable.

The Role of Variables in Algebra

Variables are symbols used to represent numbers in algebraic expressions and equations. They function as placeholders that can vary or be manipulated to help solve problems. In the context of our shower head example, we use variables to set up our equation that models the relationship between time and gallons of water used.

"\( m \)" is our variable for minutes, representing the time that the shower runs. This variable is crucial because it allows us to plug in different minute values freely to see how many gallons are used over time. "\( g \)" stands for gallons, the outcome dependent on the number of minutes we input.

Assigning meaningful variables not only helps set up the initial equation but also makes it easier for anyone reading or using your equation to understand what each part represents. This clarity lets you manipulate and understand the relationship between different quantities.

Exploring the Rate of Change

The "rate of change" in a linear equation describes how one quantity changes concerning another. It essentially highlights the constant relationship between two variables over a given time or interval, acting similar to the slope in these equations.

In the equation \( g = 6m \), the rate of change is the number 6. This number shows how the gallons of water used change with each additional minute. In other words, for each minute that passes, the gallons increase by 6. This consistency brings predictability to the equation, making it easier to graph or compute various scenarios quickly.

Understanding the rate of change can also assist in assessing the efficiency of processes. Knowing how quickly or slowly something happens over time can influence decisions in budgeting resources, like selecting a more efficient shower head if saving water is the goal.