Unit I Worksheet 1: Graphing Practice - Students analyze data sets to determine mathematical expressions by graphing and interpreting relationships.

Problem Overview:

The task involves analyzing four data sets, graphing the original data, identifying the relationship between variables, and modifying the data to plot as a straight line. From the linear fit of the modified data, we determine the mathematical expression for the relationship between the variables.

Data Set 1:

#### Original Data:
| \( V \) (m³) | \( P \) (Pa) |
|--------------|-------------|
| 0.1 | 40 |
| 0.5 | 8 |
| 1 | 4 |
| 2 | 2 |
| 4 | 1 |
| 5 | 0.8 |
| 8 | 0.5 |
| 10 | 0.4 |

#### Graph and Analysis:
- The graph shows an inverse relationship between \( V \) and \( P \), i.e., \( P \propto \frac{1}{V} \).
- To linearize the data, we plot \( P \) vs. \( \frac{1}{V} \).

#### Modified Data:
| \( \frac{1}{V} \) (m⁻³) | \( P \) (Pa) |
|-------------------------|-------------|
| 10 | 40 |
| 2 | 8 |
| 1 | 4 |
| 0.5 | 2 |
| 0.25 | 1 |
| 0.2 | 0.8 |
| 0.125 | 0.5 |
| 0.1 | 0.4 |

#### Linear Fit:
- The modified graph is a straight line.
- The slope \( m \) of the line is the constant of proportionality.
- From the data, the relationship is \( P = k \cdot \frac{1}{V} \), where \( k \) is the slope.

#### Mathematical Expression:
\[ P = \frac{4}{V} \]
or
\[ PV = 4 \]

Data Set 2:

#### Original Data:
| \( t \) (s) | \( x \) (m) |
|-------------|-------------|
| 0.1 | 0.03 |
| 0.2 | 0.12 |
| 0.5 | 0.75 |
| 1 | 3 |
| 2 | 12 |
| 3 | 27 |
| 4 | 48 |
| 5 | 75 |

#### Graph and Analysis:
- The graph shows a parabolic relationship between \( t \) and \( x \), i.e., \( x \propto t^2 \).
- To linearize the data, we plot \( x \) vs. \( t^2 \).

#### Modified Data:
| \( t^2 \) (s²) | \( x \) (m) |
|---------------|-------------|
| 0.01 | 0.03 |
| 0.04 | 0.12 |
| 0.25 | 0.75 |
| 1 | 3 |
| 4 | 12 |
| 9 | 27 |
| 16 | 48 |
| 25 | 75 |

#### Linear Fit:
- The modified graph is a straight line.
- The slope \( m \) of the line is the constant of proportionality.
- From the data, the relationship is \( x = k \cdot t^2 \), where \( k \) is the slope.

#### Mathematical Expression:
\[ x = 3t^2 \]

Data Set 3:

#### Original Data:
| \( A \) (months) | \( W \) (kg) |
|------------------|--------------|
| 1 | 7.2 |
| 2 | 9.4 |
| 3 | 10.5 |
| 4 | 12.0 |
| 5 | 13.0 |
| 6 | 14.3 |
| 7 | 15.2 |
| 8 | 16.7 |

#### Graph and Analysis:
- The graph shows a linear relationship between \( A \) and \( W \).
- No modification is needed since the data is already linear.

#### Linear Fit:
- The slope \( m \) and y-intercept \( b \) can be determined from the data.
- The relationship is \( W = mA + b \).

#### Mathematical Expression:
\[ W = 1.2A + 6 \]

Data Set 4:

#### Original Data:
| \( t \) (s) | \( v \) (mm/s) |
|-------------|----------------|
| 1 | 10 |
| 1.2 | 20 |
| 2.7 | 30 |
| 4.8 | 40 |
| 7.5 | 50 |
| 10.8 | 60 |
| 14.7 | 70 |
| 19.2 | 80 |

#### Graph and Analysis:
- The graph shows a square root relationship between \( t \) and \( v \), i.e., \( v \propto \sqrt{t} \).
- To linearize the data, we plot \( v \) vs. \( \sqrt{t} \).

#### Modified Data:
| \( \sqrt{t} \) (s¹/²) | \( v \) (mm/s) |
|----------------------|----------------|
| 1 | 10 |
| 1.1 | 20 |
| 1.6 | 30 |
| 2.2 | 40 |
| 2.7 | 50 |
| 3.3 | 60 |
| 3.8 | 70 |
| 4.4 | 80 |

#### Linear Fit:
- The modified graph is a straight line.
- The slope \( m \) of the line is the constant of proportionality.
- From the data, the relationship is \( v = k \cdot \sqrt{t} \), where \( k \) is the slope.

#### Mathematical Expression:
\[ v = 10\sqrt{t} \]

Final Answers:

1. \( P = \frac{4}{V} \)
2. \( x = 3t^2 \)
3. \( W = 1.2A + 6 \)
4. \( v = 10\sqrt{t} \)

\[
\boxed{
P = \frac{4}{V}, \quad x = 3t^2, \quad W = 1.2A + 6, \quad v = 10\sqrt{t}
}
\]