Answer :
Let's analyze the given data step-by-step to determine which statements are true about the relationship between the depth of snow and the height of water in the jar after the snow melts.
Here is the provided table:
[tex]\[ \begin{array}{|c|c|} \hline \text{Snow Depth (in.), } x & \text{Water Depth (in.), } f(x) \\ \hline 2 & 0.4 \\ \hline 4 & 0.8 \\ \hline 6 & 1.2 \\ \hline 8 & 1.6 \\ \hline 10 & 2.0 \\ \hline \end{array} \][/tex]
### Analyzing the Data
#### Step 1: Determine the relationship between snow depth and water depth.
- Calculate the ratio of water depth to snow depth for each pair of values:
[tex]\[ \frac{f(2)}{2} = \frac{0.4}{2} = 0.2, \quad \frac{f(4)}{4} = \frac{0.8}{4} = 0.2, \quad \frac{f(6)}{6} = \frac{1.2}{6} = 0.2, \quad \frac{f(8)}{8} = \frac{1.6}{8} = 0.2, \quad \frac{f(10)}{10} = \frac{2.0}{10} = 0.2 \][/tex]
The ratio is consistent for all data points, indicating a linear relationship of the form [tex]\( f(x) = 0.2x \)[/tex].
#### Step 2: Evaluate the given statements based on the linear relationship.
Statement 1: The points on a graph representing the relationship lie on a line.
- The relationship is linear ([tex]\( f(x) = 0.2x \)[/tex]), so this statement is true.
Statement 2: There is 0.4 inch of water to every 1 inch of snow.
- The correct ratio is 0.2 inches of water to 1 inch of snow, so this statement is false.
Statement 3: A line through the points will pass through [tex]\((0,0)\)[/tex].
- The function [tex]\( f(x) = 0.2x \)[/tex] indicates that when [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex]. Hence, the line passes through (0,0). This statement is true.
Statement 4: The function relating snow depth to water depth is quadratic.
- The given data indicates a linear relationship, not a quadratic one. So, this statement is false.
Statement 5: The data can be represented by [tex]\( f(x) = 0.2^x \)[/tex].
- The data represents a linear relationship, not an exponential one. [tex]\( f(x) = 0.2x \)[/tex] is the correct form, making this statement false.
### Conclusion:
The valid statements based on the analysis are:
1. The points on a graph representing the relationship lie on a line.
3. A line through the points will pass through [tex]\((0,0)\)[/tex].
Therefore, the correct answers are:
- The points on a graph representing the relationship lie on a line.
- A line through the points will pass through [tex]\((0,0)\)[/tex].
Here is the provided table:
[tex]\[ \begin{array}{|c|c|} \hline \text{Snow Depth (in.), } x & \text{Water Depth (in.), } f(x) \\ \hline 2 & 0.4 \\ \hline 4 & 0.8 \\ \hline 6 & 1.2 \\ \hline 8 & 1.6 \\ \hline 10 & 2.0 \\ \hline \end{array} \][/tex]
### Analyzing the Data
#### Step 1: Determine the relationship between snow depth and water depth.
- Calculate the ratio of water depth to snow depth for each pair of values:
[tex]\[ \frac{f(2)}{2} = \frac{0.4}{2} = 0.2, \quad \frac{f(4)}{4} = \frac{0.8}{4} = 0.2, \quad \frac{f(6)}{6} = \frac{1.2}{6} = 0.2, \quad \frac{f(8)}{8} = \frac{1.6}{8} = 0.2, \quad \frac{f(10)}{10} = \frac{2.0}{10} = 0.2 \][/tex]
The ratio is consistent for all data points, indicating a linear relationship of the form [tex]\( f(x) = 0.2x \)[/tex].
#### Step 2: Evaluate the given statements based on the linear relationship.
Statement 1: The points on a graph representing the relationship lie on a line.
- The relationship is linear ([tex]\( f(x) = 0.2x \)[/tex]), so this statement is true.
Statement 2: There is 0.4 inch of water to every 1 inch of snow.
- The correct ratio is 0.2 inches of water to 1 inch of snow, so this statement is false.
Statement 3: A line through the points will pass through [tex]\((0,0)\)[/tex].
- The function [tex]\( f(x) = 0.2x \)[/tex] indicates that when [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0 \)[/tex]. Hence, the line passes through (0,0). This statement is true.
Statement 4: The function relating snow depth to water depth is quadratic.
- The given data indicates a linear relationship, not a quadratic one. So, this statement is false.
Statement 5: The data can be represented by [tex]\( f(x) = 0.2^x \)[/tex].
- The data represents a linear relationship, not an exponential one. [tex]\( f(x) = 0.2x \)[/tex] is the correct form, making this statement false.
### Conclusion:
The valid statements based on the analysis are:
1. The points on a graph representing the relationship lie on a line.
3. A line through the points will pass through [tex]\((0,0)\)[/tex].
Therefore, the correct answers are:
- The points on a graph representing the relationship lie on a line.
- A line through the points will pass through [tex]\((0,0)\)[/tex].
