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Student Math Lab: Rainwater Catchment Geometry

Applied Geometry, Volume, & Unit Conversions
Objective for the Student: Calculate the real-world water harvesting potential of a residential roof structure and evaluate container volume constraints to ensure zero resource overflow.

Core Conversion Constants to Use:

Part 1: Calculating the Footprint and Yield

Problem 1: Determining the Catchment Area Footprint

A house has a rectangular roof footprint. When measured from the ground looking directly downward, the main structure has a length of $40\text{ feet}$ and a width of $30\text{ feet}$. Additionally, an attached porch roof extends the footprint by an extra $12\text{ feet}$ by $15\text{ feet}$.

Calculate the total flat horizontal catchment footprint area ($A$) in square feet ($ft^2$).

Note: Do not calculate the slanted slope area of the roof. Rain falls vertically, so only the horizontal flat structural footprint maps the true interception area!

Problem 2: Total Volume in Cubic Feet

During a heavy summer storm, a weather station measures exactly $1.5\text{ inches}$ of rainfall. Convert this depth from inches into feet, and then compute the raw volume ($V_{raw}$) of water in cubic feet ($ft^3$) that fell directly over the total roof area ($A$) from Problem 1.

Formula Hint: $V = \text{Area (}ft^2\text{)} \times \text{Depth (}ft\text{)}$

Problem 3: The Realized Gallon Output

Now, let's translate that abstract volume into usable household units.

  1. Convert your raw cubic feet volume ($V_{raw}$) from Problem 2 into fluid gallons.
  2. Apply the **90% collection efficiency coefficient** ($E_c = 0.90$) to find the true amount of net harvestable gallons ($G_{net}$) saved by the gutters. Round your final answer to the nearest whole gallon.

Part 2: Engineering the Storage Container Constraints

Problem 4: Cistern Volume Architecture

The family wants to store this water in a vertical cylindrical tank (cistern) located next to the house. The tank has a circular baseline radius of $r = 4\text{ feet}$ and a total internal height of $h = 8\text{ feet}$.

Calculate the total holding volume of this cylinder in cubic feet, and then convert that volume into max container capacity in **gallons**. (Use $\pi \approx 3.1416$, and round to the nearest whole gallon).

Cylinder Volume Formula: $V_{tank} = \pi \cdot r^2 \cdot h$

Problem 5: Managing the Overflow Variable

Assume the tank was already partially full, holding exactly $1,200\text{ gallons}$ of water before the storm began.

  1. Add the new net harvestable storm water ($G_{net}$) from Problem 3 to the starting water volume. Does the combined water exceed the max container capacity calculated in Problem 4?
  2. If it overflows, calculate exactly how many gallons of water will spill over. If it doesn't overflow, calculate how many gallons of extra space are left in the tank.

Parent & Educator Evaluation Matrix (Answer Key)

Teachers/Parents: Use this section to review your student's calculation matrices and logical steps.

Problem 1 Answer:
$$\text{Main House Area: } 40\text{ ft} \times 30\text{ ft} = 1,200\text{ ft}^2$$ $$\text{Porch Extension Area: } 12\text{ ft} \times 15\text{ ft} = 180\text{ ft}^2$$ $$\text{Total Area (}A\text{): } 1,200 + 180 = \mathbf{1,380\text{ ft}^2}$$
Problem 2 Answer:
$$\text{Convert Depth: } \frac{1.5\text{ inches}}{12\text{ inches/ft}} = 0.125\text{ feet of rain depth.}$$ $$\text{Volume (}V_{raw}\text{): } 1,380\text{ ft}^2 \times 0.125\text{ ft} = \mathbf{172.5\text{ ft}^3}$$
Problem 3 Answer:
Step 1 (Raw Gallon Conversion): $$172.5\text{ ft}^3 \times 7.48\text{ gallons/ft}^3 = 1,290.3\text{ raw gallons}$$ Step 2 (Apply Efficiency Variable): $$G_{net} = 1,290.3 \times 0.90 = 1,161.27 \rightarrow \mathbf{1,161\text{ Net Harvested Gallons}}$$

Quick Calculation Shortcut Note: Combining constants ($\frac{1}{12} \times 7.48 \times 0.90$) yields approximately **$0.561$ gallons per square foot per inch of rain**. $1,380 \times 1.5 \times 0.561 \approx 1,161$ gallons.

Problem 4 Answer:
$$V_{tank} = 3.1416 \times (4)^2 \times 8 = 3.1416 \times 16 \times 8 = 402.125\text{ ft}^3$$ $$\text{Capacity in Gallons: } 402.125 \times 7.48 = 3,007.89 \rightarrow \mathbf{3,008\text{ Gallons Maximum Capacity}}$$
Problem 5 Answer:
$$\text{Total System Load: } 1,200\text{ starting gallons} + 1,161\text{ storm gallons} = 2,361\text{ total gallons.}$$
  1. No, it does not overflow. The total system load ($2,361\text{ gal}$) is less than the max capacity limit ($3,008\text{ gal}$).
  2. Remaining Margin Space: $3,008 - 2,361 = \mathbf{647\text{ gallons of extra volume capacity remaining}}$ inside the cistern tank.