AMC 10 · 2017 · #14
Grade 8 geometry-3dPick an answer.
There is no formula for 'ice cream novelty item', so I break the solid into pieces I do have formulas for. The upper piece is already a plain cone. The lower piece, the frustum, is not — but a frustum is a big cone with a small cone removed, so I can replace one hard shape with two easy ones. To find that removed tip I slice the solid down its axis and look at the flat picture: the slanted side is a straight line whose radius changes at a steady rate, and following that line down tells me exactly where the missing apex sits. Then the whole problem is three cone volumes combined.
Slice it down the middle
A vertical slice shows how the radius changes.
A shape spun around an axis is fully described by the flat outline you get when you cut it open, so a 3D question becomes a 2D picture.
7.G.A.3Draw A DiagramSplit into two pieces
The total splits into two pieces.
Volume is additive: if two solids share only a flat face, filling each one separately fills the whole thing exactly once.
7.G.B.6Identify SubproblemsVolume of the top cone
The top cone goes straight into the formula.
A cone holds exactly one third of the cylinder that would wrap around it with the same base and height.
A cone holds exactly one third of the cylinder that would wrap around it with the same base and height.
▸ Why?
A solid that tapers evenly to a point fills one third of the straight solid on the same base and height.
▸ Why?
That straight solid is its base area repeated all the way up, so the two volumes share one base and one height.
Find the frustum's missing tip
Extending the slope locates the missing tip.
The cut-off tip is a scaled-down copy of the whole cone, so its radius and its height shrink by the same factor.
8.G.A.4Visualize Spatial RelationshipsFrustum = big cone minus small cone
The frustum is the big cone minus the small one.
Putting the missing tip back turns an awkward shape into two shapes you already know, and subtraction removes the part you added.
8.G.C.9Solve An Easier Related ProblemAdd the two volumes
Adding gives forty-four pi over three.
Once every piece is measured in the same units, the total is just the sum.
7.G.B.6Identify SubproblemsWhen a solid has no formula, put back the piece that was cut off: a frustum is just a big cone minus the small cone that used to be its tip.
- Slice it down the middle
- Split into two pieces
- Volume of the top cone
- Find the frustum's missing tip
- Frustum = big cone minus small cone
- Add the two volumes
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