Problem 28 - Algebra 2

Question

A woman painting a room will burn an average of 4.5 Calories per minute. Write an equation for the number of Calories burned in \(m\) minutes.

Step-by-Step Solution

Answer

The equation is \( C = 4.5m \).

Step 1: Identify the Variables

We need to establish the variables involved in this scenario. Here, the variable is the number of minutes, which we'll represent as \( m \). The Calories burned per minute is a constant value provided in the problem, which is 4.5 Calories per minute.

Step 2: Set Up the Equation

To find the total Calories burned, we multiply the number of minutes \( m \) by the Calories burned per minute (4.5). Hence, the equation becomes \( C = 4.5m \), where \( C \) represents the total Calories burned.

Step 3: Equation Verification

To ensure accuracy, consider a sample calculation. For example, if \( m = 10 \) (minutes), substituting into the equation gives \( C = 4.5 \times 10 = 45 \) Calories, which is consistent with the provided rate of burning.

Key Concepts

Calories Burned

When we talk about `calories burned`, we refer to the amount of energy expended during a physical activity. This is useful for understanding how much work our bodies are doing. For instance, painting a room can be surprisingly energetic. In the example above, the woman burns 4.5 Calories every minute she paints. Understanding this concept is important for a few reasons:

• It helps in planning and monitoring workout goals.
• You can better manage daily calorie intake and expenditure, which is key to maintaining a healthy lifestyle.

Each activity has a different calorie-burning rate; knowing these helps in choosing activities that align with fitness goals. Additionally, such calculations are crucial for those trying to create a calorie deficit for weight loss.

Rate of Change

The `rate of change` describes how one quantity changes in relation to another. In the context of our exercise, it refers to how the calories burned increase with each passing minute while painting. Here, the rate of change is a constant 4.5 Calories per minute, indicating a linear relationship between time spent painting and calories burned. This linear relationship means:

• For every additional minute, the number of calories burned increases uniformly by 4.5.
• The graph of this relationship would be a straight line with a positive slope, showing continuous and predictable growth.

Understanding the rate of change can clarify how spending more time in an activity leads to more calories being burned, making it an essential component in fitness calculations.

Variables in Equations

Equations often use `variables` to represent changing values. In our example, the letter `m` is used as a variable to denote the number of minutes. The importance of variables lies in their ability to:

• Make equations general and flexible, applicable to various scenarios.
• Allow mathematical modeling of dynamic situations where inputs can change.

We also have constants, like the number 4.5 representing calories burned each minute, which do not change. An equation like this one helps in expressing a real-world scenario mathematically, enabling predictions based on different values of `m` and helping us understand the relationship between the time spent and calories burned. By mastering how to handle variables, you can tackle a wide range of real-life problems using mathematics.