Answer :
To address the problem, let's examine each statement and determine its validity based on the given data:
1. The points on a graph representing the relationship lie on a line.
True or False?
We need to check if the relationship between snow depth [tex]\(x\)[/tex] and water depth [tex]\(f(x)\)[/tex] is linear. If we calculate the slope [tex]\(m\)[/tex] for each pair of points, it should be constant for the relationship to be linear.
The slope [tex]\(m\)[/tex] is calculated as follows:
[tex]\[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
For the given data:
- Between (2, 0.4) and (4, 0.8):
[tex]\[ m = \frac{0.8 - 0.4}{4 - 2} = \frac{0.4}{2} = 0.2 \][/tex]
- Between (4, 0.8) and (6, 1.2):
[tex]\[ m = \frac{1.2 - 0.8}{6 - 4} = \frac{0.4}{2} = 0.2 \][/tex]
- Between (6, 1.2) and (8, 1.6):
[tex]\[ m = \frac{1.6 - 1.2}{8 - 6} = \frac{0.4}{2} = 0.2 \][/tex]
- Between (8, 1.6) and (10, 2.0):
[tex]\[ m = \frac{2.0 - 1.6}{10 - 8} = \frac{0.4}{2} = 0.2 \][/tex]
Since the slope [tex]\(m\)[/tex] is constant at 0.2 for all pairs of points, the points do indeed lie on a line.
Result: True
2. There is 0.4 inch of water to every 1 inch of snow.
True or False?
To assess this statement, we need to see if the ratio of water depth to snow depth consistently equals 0.4 per inch.
Examining the given data, we calculate the ratio for each:
[tex]\[ \frac{0.4}{2} = 0.2, \quad \frac{0.8}{4} = 0.2, \quad \frac{1.2}{6} = 0.2, \quad \frac{1.6}{8} = 0.2, \quad \frac{2.0}{10} = 0.2 \][/tex]
The ratio of water depth to snow depth is 0.2 (not 0.4). Thus, the statement is incorrect.
Result: False
3. A line through the points will pass through [tex]\((0,0)\)[/tex].
True or False?
If the line passes through [tex]\((0,0)\)[/tex], the relationship can be represented by [tex]\( y = mx + b \)[/tex] with [tex]\( b = 0 \)[/tex].
Given that the slope [tex]\(m\)[/tex] has been established as 0.2 and we can write the linear equation as [tex]\( y = 0.2x \)[/tex]. When [tex]\(x = 0\)[/tex], [tex]\(y = 0.2 \cdot 0 = 0\)[/tex], showing that the line does indeed pass through the origin.
Result: True
4. The function relating snow depth to water depth is quadratic.
True or False?
A quadratic function has the form [tex]\( f(x) = ax^2 + bx + c \)[/tex]. Based on the data given, the relationship is linear, as shown by the constant slope and linear equation [tex]\( y = 0.2x \)[/tex].
Result: False
5. The data can be represented by [tex]\( f(x) = 0.2^x \)[/tex].
True or False?
The function [tex]\( f(x) = 0.2^x \)[/tex] represents an exponential relationship. Given the data, the relationship is linear, not exponential.
Result: False
Thus, the correct statements based on the given data 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].
1. The points on a graph representing the relationship lie on a line.
True or False?
We need to check if the relationship between snow depth [tex]\(x\)[/tex] and water depth [tex]\(f(x)\)[/tex] is linear. If we calculate the slope [tex]\(m\)[/tex] for each pair of points, it should be constant for the relationship to be linear.
The slope [tex]\(m\)[/tex] is calculated as follows:
[tex]\[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
For the given data:
- Between (2, 0.4) and (4, 0.8):
[tex]\[ m = \frac{0.8 - 0.4}{4 - 2} = \frac{0.4}{2} = 0.2 \][/tex]
- Between (4, 0.8) and (6, 1.2):
[tex]\[ m = \frac{1.2 - 0.8}{6 - 4} = \frac{0.4}{2} = 0.2 \][/tex]
- Between (6, 1.2) and (8, 1.6):
[tex]\[ m = \frac{1.6 - 1.2}{8 - 6} = \frac{0.4}{2} = 0.2 \][/tex]
- Between (8, 1.6) and (10, 2.0):
[tex]\[ m = \frac{2.0 - 1.6}{10 - 8} = \frac{0.4}{2} = 0.2 \][/tex]
Since the slope [tex]\(m\)[/tex] is constant at 0.2 for all pairs of points, the points do indeed lie on a line.
Result: True
2. There is 0.4 inch of water to every 1 inch of snow.
True or False?
To assess this statement, we need to see if the ratio of water depth to snow depth consistently equals 0.4 per inch.
Examining the given data, we calculate the ratio for each:
[tex]\[ \frac{0.4}{2} = 0.2, \quad \frac{0.8}{4} = 0.2, \quad \frac{1.2}{6} = 0.2, \quad \frac{1.6}{8} = 0.2, \quad \frac{2.0}{10} = 0.2 \][/tex]
The ratio of water depth to snow depth is 0.2 (not 0.4). Thus, the statement is incorrect.
Result: False
3. A line through the points will pass through [tex]\((0,0)\)[/tex].
True or False?
If the line passes through [tex]\((0,0)\)[/tex], the relationship can be represented by [tex]\( y = mx + b \)[/tex] with [tex]\( b = 0 \)[/tex].
Given that the slope [tex]\(m\)[/tex] has been established as 0.2 and we can write the linear equation as [tex]\( y = 0.2x \)[/tex]. When [tex]\(x = 0\)[/tex], [tex]\(y = 0.2 \cdot 0 = 0\)[/tex], showing that the line does indeed pass through the origin.
Result: True
4. The function relating snow depth to water depth is quadratic.
True or False?
A quadratic function has the form [tex]\( f(x) = ax^2 + bx + c \)[/tex]. Based on the data given, the relationship is linear, as shown by the constant slope and linear equation [tex]\( y = 0.2x \)[/tex].
Result: False
5. The data can be represented by [tex]\( f(x) = 0.2^x \)[/tex].
True or False?
The function [tex]\( f(x) = 0.2^x \)[/tex] represents an exponential relationship. Given the data, the relationship is linear, not exponential.
Result: False
Thus, the correct statements based on the given data 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].
