Problem 100 - College Algebra with Modeling and Visualization

Question

Going Green The average top-loading washing machine uses about 40 gallons of water per load of clothes. Write the formula for a function \(W\) that calculates the number of gallons of water used while washing \(x\) loads of clothes. Evaluate \(W(30)\) and interpret your result.

Step-by-Step Solution

Answer

For 30 loads, 1200 gallons are used, indicating high water usage.

Step 1: Define the Function Formula

To find the formula for the function \( W \), recognize that each load of clothes uses 40 gallons of water. Therefore, the function that represents the gallons of water used for \( x \) loads is: \[ W(x) = 40x \] This formula simply multiplies the number of loads, \( x \), by 40 to calculate the total water usage.

Step 2: Substitute into the Function

Now, substitute 30 into the function \( W \) to find out how many gallons are used for 30 loads. Using the formula from the previous step:\[ W(30) = 40 \times 30 \]

Step 3: Calculate the Result

Perform the multiplication:\[ W(30) = 40 \times 30 = 1200 \]So, 30 loads of clothes use 1200 gallons of water.

Step 4: Interpret the Result

The evaluation of \( W(30) = 1200 \) means that washing 30 loads of clothes will consume 1200 gallons of water. This indicates substantial water usage and hints at the environmental impact of laundry activities.

Key Concepts

Linear Functions

Linear functions are mathematical tools used to describe a direct relationship between two variables. In our exercise, the linear function is represented by the equation \( W(x) = 40x \). This notation tells us that the number of gallons of water used is directly proportional to the number of laundry loads. Here’s how they work:

• The equation is in the form of \( y = mx + b \). In \( W(x) = 40x \), \( m \), the slope, is 40, which shows the rate of water usage per load. There is no \( b \), which means the line passes through the origin.
• Because there is a consistent rate per load, this creates a straight line on a graph.
• This is useful for predicting water usage for any number of laundry loads by simply substituting values for \( x \).

Understanding linear functions helps us calculate outcomes based on a consistent relationship, like how much water is used in doing laundry.

Environmental Impact

The environmental impact represents the effect that activities have on the natural world. Laundry, for example, has a significant environmental footprint due to water usage. In our example with a top-loading washing machine, using 1200 gallons for 30 loads highlights:

• **Resource Consumption**: Water is a finite resource. Using large quantities for daily activities strains it.
• **Energy Use**: Water requires energy to pump and heat. Thus, high water usage also leads to increased energy consumption.
• **Pollution**: Wastewater from washing machines can carry detergents and chemicals into water systems, impacting aquatic life.

By understanding the function \( W(x) = 40x \), we become more aware of the extensive resources involved in common tasks. This awareness is a crucial step toward making more sustainable choices.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or relationship. In the problem, \( W(x) = 40x \) is an algebraic expression used to calculate water consumption per load:

• **Variables and Constants**: \( x \) is the variable representing the number of loads, while 40 is a constant, signifying the gallons per load.
• **Simplifying Calculations**: By multiplying these two, you determine total water usage simply and consistently.
• **Evaluation**: Plugging different numbers into the expression helps evaluate various scenarios like \( W(30) = 1200 \) gallons.

Algebraic expressions are powerful in translating everyday problems into mathematical language, making complex calculations straightforward.