Integrated Math 3 / Unit 7: Geometric Modeling
7.6 Modeling with Geometric Shapes
Notes
Break a real object into shapes you know, then add or subtract.
- Composite solids: add the volumes. A silo is a cylinder plus a hemisphere, where a hemisphere has volume $\displaystyle \frac{2}{3}\pi r^3$.
- Design constraints: use a volume formula to solve for a missing dimension.
- Scale models: if lengths scale by $k$, then areas scale by $k^2$ and volumes by $k^3$.
Practice
A silo is a cylinder with radius 3 and height 10 topped by a hemisphere with radius 3. What is its volume?
Cylinder: $\pi(9)(10) = 90\pi$. Hemisphere: $\displaystyle \frac{2}{3}\pi(27) = 18\pi$. Total: $108\pi$.
A can must hold $400 \text{ cm}^3$ and have a radius of 4 cm. About how tall must it be?
$400 = \pi(16)h$, so $\displaystyle h = \frac{400}{16\pi} \approx 7.96$ cm.
A model ship is built at a scale of 1 to 50. The real ship's volume is how many times the model's volume?
Volumes scale by $50^3 = 125{,}000$.
Ready for more? Take the Unit 7 test.