Comprehensive Study Guide on Linear, Area, and Volume Scale Factors

Core Definitions and Dimensionality:
- Linear scale factor ( $k$ ): Represents the ratio between corresponding one-dimensional measurements (such as height, length, width, radius, or perimeter) of a scaled model or copy and its original object.
- Area scale factor ( $k^2$ ): Represents the ratio between corresponding two-dimensional surface areas, cross-sectional areas, or sail areas of a scaled model and its original object.
- Volume scale factor ( $k^3$ ): Represents the ratio between corresponding three-dimensional volume or capacity measurements of a scaled model and its original object.
Formulaic Relationships:
- Linear calculation: $\text{Original Linear Dimension} \times k = \text{New Linear Dimension}$
- Area calculation: $\text{Original Area} \times k^2 = \text{New Area}$
- Volume calculation: $\text{Original Volume} \times k^3 = \text{New Volume}$
- Scale factor ratio: $k = \frac{\text{New Dimension}}{\text{Original Dimension}}$
Introductory Mathematical Verification:
- Whiteboard dimension check: $9 \times 5.125 = 46.125$
- Extended calculation: $12 \times 5.125 = 61.5$
- Area verification check: $61.5 \times 46.125$ confirms consistency with the original area measurement.

Model Ship Given Parameters:
- Scale ratio (New to Old / Model to Original): $1 : 30$ , which establishes $k = \frac{1}{30}$ .
- Model height (Linear dimension): $16\,\text{cm}$ .
- Model sail area (Area dimension): $8.5\,\text{cm}^2$
Determining Actual Ship Height (Linear Calculation):
- Let $x$ represent the height of the original (actual) ship.
- Setting up the equation: $x \times k = \text{Model Height}$
- Substitution: $x \times \left(\frac{1}{30}\right) = 16\,\text{cm}$
- Solving for $x$ by dividing both sides by $\frac{1}{30}$ (or multiplying by the reciprocal $30$ ):
- $x = 16\,\text{cm} \times 30$
- $x = 480\,\text{cm}$
- Conversion to meters: $480\,\text{cm} = 4.8\,\text{m}$
- Alternative setup using cross-multiplication:
- $\frac{1}{30} = \frac{16\,\text{cm}}{x}$
- $x = 16\,\text{cm} \times 30 = 480\,\text{cm}$
Determining Actual Sail Area (Area Calculation):
- Let $y$ represent the sail area of the original (actual) ship.
- Setting up the area formula: $y \times k^2 = \text{Model Sail Area}$
- Calculating $k^2$ : $k^2 = \left(\frac{1}{30}\right)^2 = \frac{1}{900}$
- Substitution: $y \times \left(\frac{1}{900}\right) = 8.5\,\text{cm}^2$
- Solving for $y$ by dividing both sides by $\frac{1}{900}$ (or multiplying by $900$ ):
- $y = 8.5\,\text{cm}^2 \times 900$
- $y = 7650\,\text{cm}^2$
- Alternative cross-multiplication setup:
- $\frac{1}{900} = \frac{8.5\,\text{cm}^2}{y}$
- $y = 8.5\,\text{cm}^2 \times 900 = 7650\,\text{cm}^2$

Storm Round Rock / Okotoks Wastewater Management System Parameters:
- Existing tank interior dimensions:
- Length: $100\,\text{m}$
- Width: $30\,\text{m}$
- Height: $50\,\text{m}$
- Capacity requirement: The new tank capacity (volume) must be $10$ times the capacity of the current tank.
Derivation of Linear Scale Factor ( $k$ ):
- The volume scale factor is given directly as $k^3 = 10$
- Taking the cube root of both sides to obtain $k$ :
- $k = \sqrt[3]{10}$
Calculating Interior Dimensions of the New Tank (Rounded to the nearest meter):
- New Length: $\text{Original Length} \times k = 100\,\text{m} \times \sqrt[3]{10} \approx 100 \times 2.1544 = 215.44\,\text{m} \approx 215\,\text{m}$
- New Width: $\text{Original Width} \times k = 30\,\text{m} \times \sqrt[3]{10} \approx 30 \times 2.1544 = 64.63\,\text{m} \approx 65\,\text{m}$
- New Height: $\text{Original Height} \times k = 50\,\text{m} \times \sqrt[3]{10} \approx 50 \times 2.1544 = 107.72\,\text{m} \approx 108\,\text{m}$

Spherical Tank Parameters:
- First (smaller) tank capacity: $1400\,\text{m}^3$
- Second (larger) tank capacity: $4725\,\text{m}^3$
Volume Scale Factor and Fill Time Calculation:
- Formula for volume scale factor: $k^3 = \frac{\text{New Capacity}}{\text{Original Capacity}}$
- Calculation: $k^3 = \frac{4725\,\text{m}^3}{1400\,\text{m}^3} = 3.375$
- Practical interpretation: The volume is $3.375$ times larger, so it takes $3.375$ times longer to fill the larger tank than the smaller tank.
Radius Scale Factor (Linear Measurement Calculation):
- Radius is a one-dimensional linear measurement (the distance from the center of the sphere to its edge).
- Formula converting volume scale factor to linear scale factor: $k = \sqrt[3]{k^3}$
- Calculation: $k = \sqrt[3]{3.375} = 1.5$
- Conclusion: The radius of the larger tank is $1.5$ times greater than the radius of the smaller tank.

Water Bottle Parameters:
- Smaller bottle height: $15\,\text{cm}$
- Smaller bottle volume: $0.7\,\text{L}$
- Larger bottle height: $45\,\text{cm}$
Determining Volume of the Larger Water Bottle:
- Method A (Assigning larger bottle as original $x$ , smaller as new):
- Linear scale factor: $k = \frac{15\,\text{cm}}{45\,\text{cm}} = \frac{1}{3}$
- Volume relationship: $x \times k^3 = \text{New Volume}$
- Substitution: $x \times \left(\frac{1}{3}\right)^3 = 0.7\,\text{L}$
- Cubing the fraction: $\left(\frac{1}{3}\right)^3 = \frac{1}{27}$
- Equation: $x \times \frac{1}{27} = 0.7\,\text{L}$
- Solving for $x$ : $x = 0.7\,\text{L} \div \left(\frac{1}{27}\right) = 0.7\,\text{L} \times 27 = 18.9\,\text{L}$
- Method B (Assigning smaller bottle as original, larger as new $y$ ):
- Linear scale factor: $k = \frac{45\,\text{cm}}{15\,\text{cm}} = 3$
- Volume relationship: $y = \text{Original Volume} \times k^3$
- Substitution: $y = 0.7\,\text{L} \times 3^3 = 0.7\,\text{L} \times 27 = 18.9\,\text{L}$
Directional Behavior of Scale Factors:
- Scaling down (large to small): Scale factor is less than $1$ ( $k < 1$ ).
- Scaling up (small to large / enlargement): Scale factor is greater than $1$ ( $k > 1$ ).
- Standard volume formula for rectangular objects: $\text{Volume} = \text{length} \times \text{width} \times \text{height}$

Book Selections and Page Lengths:
- Brinley selected The Snow Child for independent reading, noting it was emotionally intense.
- Invisible Man was considered but passed over due to its length of approximately $600$ pages.
- Metamorphosis was noted for being short ( $50$ pages long) and dealing with freaky psychological concepts.
- Carry On was also mentioned as a reading choice.
Details and Plot Points of The Snow Child:
- Based on a fairy tale/fable about two old people (not old men) who cannot have children, build a snowman, and it turns into a snow child or spirit.
- In traditional variations of the tale, the snow child dies in approximately $10$ different ways.
- Plot elements discussed: A $17$ -year-old female character hangs out with a boy, experiences an unexpected pregnancy, gives birth, and dies. Currently reached page $99$ of the text.
Scheduling Discussion:
- A class meeting regarding the literature books was slated for September, but September concluded without the meeting taking place.