Problem 61 - Elementary and Intermediate Algebra

Question

The number of calories that a 125-pound adult burns doing general house cleaning chores is three times the number of minutes spent cleaning. a. Write a verbal model using the word product that describes the relationship between calories burned and minutes cleaning. b. Write a formula using the variables \(c\) and \(m\) that describes the relationship between calories burned and minutes cleaning. c. Use your answer to part b to complete the following table. \(\begin{array}{|c|c|c|c|c|c|c|}\hline m & {10} & {20} & {30} & {40} & {50} & {60} \\ \hline c & {} & {} & {} & {} & {} & {} \\ \hline\end{array}\) d. Use the data from the table to construct a line graph. Scale the horizontal axis in units of 10 minutes. Scale the vertical axis in units of 30 calories.

Step-by-Step Solution

Answer

a. "Calories burned is the product of three and minutes cleaning." b. \(c = 3m\). c. \(c\) values are 30, 60, 90, 120, 150, 180. d. Plot points (10, 30) to (60, 180), draw a line.

Step 1: Creating the Verbal Model

To describe the relationship between calories burned and minutes spent cleaning, we use the key concept that calories are calculated as a product. In this case, the number of calories burned is equal to three times the number of minutes spent cleaning. Therefore, the verbal model is: "The number of calories burned is the product of three and the number of minutes cleaning."

Step 2: Writing the Formula

Using the verbal model, we'll write a formula with two variables: let \(c\) represent the number of calories burned, and \(m\) the number of minutes spent cleaning. The relationship is expressed as \(c = 3m\).

Step 3: Completing the Table

We will use the formula \(c = 3m\) to fill in the table for each given value of \(m\).- For \(m = 10\), \(c = 3 \times 10 = 30\).- For \(m = 20\), \(c = 3 \times 20 = 60\).- For \(m = 30\), \(c = 3 \times 30 = 90\).- For \(m = 40\), \(c = 3 \times 40 = 120\).- For \(m = 50\), \(c = 3 \times 50 = 150\).- For \(m = 60\), \(c = 3 \times 60 = 180\).The completed table is:\[\begin{array}{|c|c|c|c|c|c|c|}\hlinem & 10 & 20 & 30 & 40 & 50 & 60 \\hlinec & 30 & 60 & 90 & 120 & 150 & 180 \\hline\end{array}\]

Step 4: Constructing the Line Graph

Using the completed table, we'll plot the data on a line graph. The horizontal axis (x-axis) represents the minutes \(m\), with a scale of 10 minutes per unit. The vertical axis (y-axis) represents the calories \(c\), with a scale of 30 calories per unit. Plot the points: (10, 30), (20, 60), (30, 90), (40, 120), (50, 150), and (60, 180). Draw a straight line through these points to visualize the relationship. The line should have a constant slope, indicating the consistent rate at which calories are burned over time.

Key Concepts

Verbal Model

A verbal model is a way to describe a mathematical relationship using words. It is particularly useful because it allows us to translate real-world situations into mathematical forms that can be easily understood and solved. In the given exercise, the relationship between the calories burned and the time spent cleaning is expressed verbally before being transformed into an algebraic expression.

In this case, the verbal model states: "The number of calories burned is the product of three and the number of minutes cleaning." This statement clearly shows that the key operation is multiplication, as indicated by the word 'product.' By recognizing such keywords, verbal models bridge the gap between word problems and mathematical equations, providing a foundational understanding that aids in solving subsequent steps.

Linear Equation

A linear equation is an algebraic equation that forms a straight line when graphed on a coordinate plane. It is represented in the form of \( ax + b = y \), where \( a \) and \( b \) are constants. In our exercise, we see a specific instance, expressed as \( c = 3m \).

- **Variables:** Here, \( c \) and \( m \) stand for calories burned and minutes cleaning, respectively.
- **Constant:** The number 3 is a constant that signifies the rate at which calories are burned per minute.

This formula is derived directly from the verbal model. By substituting specific values for \( m \), we can quickly calculate \( c \), which is vital for creating tables or graphs. The simplicity of a linear equation makes it an essential tool in algebraic modeling because it offers both clarity and quick interpretability. For example, knowing \( c = 3m \) helps us fill in missing values in a table and understand broader trends.

Graphing Linear Functions

Graphing a linear function involves plotting points on a graph and drawing a line through them to visually represent the equation. In this exercise, using the table obtained from the linear equation \( c = 3m \), each point \((m, c)\) is plotted to form a straight line, reflecting the constant rate of change.

- **Axes:** The horizontal axis (x-axis) is used for minutes \( m \), and the vertical axis (y-axis) is for calories \( c \). In this scenario, the x-axis increment is 10 minutes per unit, and the y-axis increment is 30 calories per unit.
- **Plotting Points:** Points like (10, 30), (20, 60) are plotted based on the table.

Once plotted, a straight line passing through these points confirms the linear relationship. This line graph illustrates the constancy and predictability of relationships in linear equations, assisting in understanding and analyzing proportional changes efficiently.

Graphing not only provides a visual dimension to algebraic expressions but also facilitates the exploration of how variables interact, thus reinforcing comprehension of algebraic modeling.