Question

Apply Graph mass versus volume for the data given in the table. What is the slope of the line? \(\begin{array}{|c|c|c|c|c|}\hline \text {Volume \)\left(\mathrm{cm}^{3}\right)\(} \ \ \ \ {7.5} & {12} & {15} & {22} \\ \text {Mass (g)} \ \ \ \ 24.1 & {38.5} & {48.0} & {70.1} \\ \hline\end{array}\)

Short answer

The slope of the line is approximately 3.17.

Step-by-step solution

Create a table of the given data

First, let's rewrite the given data in a more readable table format. You should be able to see the volume in one column, and the corresponding mass in the other column: $\begin{array}{|c|c|}\hline \text {Volume \(\left(\mathrm{cm}^{3}\right)\)} & \text{Mass (g)} \\\ \hline {7.5} & 24.1 \\ \hline {12} & {38.5} \\\ \hline {15} & {48.0} \\\ \hline {22} & {70.1} \\\ \hline\end{array}$

Plot the points on a graph

With the data in a table format, we can now plot these points on a graph. Label the x-axis as "Volume (\(\mathrm{cm}^{3}\))" and the y-axis as "Mass (g)". Plot the four points given in the table: 1. (7.5, 24.1) 2. (12, 38.5) 3. (15, 48) 4. (22, 70.1)

Draw a line of best fit

Once the points are plotted on the graph, you can now draw a line of best fit. A line of best fit is a straight line that best represents the data on the scatter plot. This line can pass through some of the points, but it's not required to pass through all of them. The goal is to create a line that represents a general trend among the data points.

Determine the slope of the line

Now that we have the line of best fit, we can find the slope of this line. To find the slope, we will use the formula: Slope = \(\frac{\text{change in y}}{\text{change in x}}\) We can select any two points on the line of best fit to determine the slope. For instance, if we choose the points (7.5, 24.1) and (22, 70.1), we can calculate the slope as follows: Slope = \(\frac{(70.1 - 24.1)}{(22 - 7.5)}\) Slope = \(\frac{46}{14.5}\) Slope = 3.17 (approximately) Therefore, the slope of the line is approximately 3.17.

Key concepts

Mass

Mass refers to the amount of matter in an object, and it is typically measured in grams or kilograms. In this exercise, each measurement of mass corresponds to the volume of the substance. By examining how mass increases with volume, you can gain insight into the density and composition of the material being studied.

• Mass can be a direct indicator of how much material is present.
• Finding a relationship between mass and volume helps to determine key characteristics like density.

This foundational concept is crucial when plotting these values on a graph because it provides a measurable means to compare against volume increases.

Volume

Volume measures the space that an object occupies, and it is expressed in cubic centimeters (cm³) in this context. When analyzing data about materials, volume allows you to understand how much space a certain amount of mass will fill.

• Different materials occupy different volumes, even if they have the same mass.
• The volume measurements serve as a point of comparison in the graphing process.

In this data plot, volume acts as the independent variable (x-axis), which helps identify patterns or trends in how mass relates to volume.

Slope

Slope is a key concept in mathematics and physics when working with graphs. It represents how steep a line is, showing the rate of change between two variables. The slope is calculated using the formula: \[ ext{Slope} = \frac{ ext{change in y}}{ ext{change in x}} \]In this exercise, the slope indicates how much mass increases per unit of volume, which ultimately reveals the density of the material.

• A steep slope means mass increases quickly with a small increase in volume.
• A gentle slope suggests a slower increase in mass with volume.

Understanding slope provides insights into the characteristics of the material across different volumes.

Graphing

Graphing is a fundamental way to visually represent data, making it easier to see relationships and trends. It involves plotting points on a two-dimensional plane using coordinates derived from the given data. For this exercise, each coordinate pair like (7.5, 24.1) or (22, 70.1) represents a point on the graph where volume is on the x-axis and mass is on the y-axis.

• The x-axis usually represents the independent variable (volume).
• The y-axis represents the dependent variable (mass).
• Graphing allows you to efficiently identify the relationship between the variables visually.

Experienced graphing helps to make sense of abstract numbers by providing a visual framework to interpret them.

Line of Best Fit

A line of best fit is a straight line that best represents the data on a scatter plot. It is crucial for identifying the underlying trend in a set of data points. Rather than connecting all points directly, this line acts as a summary of the data trend.

• The line does not need to touch all data points, but it should be as close as possible to the majority.
• It minimizes the distance between the line and each point.
• This line helps in estimating values not directly measured in the dataset.

In this exercise, the line of best fit helps calculate the slope, giving insight into how mass and volume interact through estimated correlations.