AMC 10 · 2021 · #6
Grade 8 geometry-3dPick an answer.
Tool #7 (Subproblems): split into (a) compute the cone's water volume, (b) compute the cylinder's base area, (c) solve volume = base area × height for the height. Tool #8 (Units): cm³ ÷ cm² = cm — confirms the formula gives a length. Tool #9 (Easier Related Problem): the cylinder's radius is exactly 2 × the cone's, so its base area is 4 × the cone's base. A pure scaling shortcut: if the radii were equal the cone-to-cylinder height ratio would be 1/3; doubling the cylinder radius further divides by 4, so the height becomes h_c/(3 · 4) = 18/12 = 1.5. Tool #3 (Eliminate): the heights below h_c / 3 = 6 are the only physically sensible candidates.
Find the cone's volume
A cone carries a factor of one third.
Grade 8 cone volume: 1/3 π r² h — same as a cylinder of the same base and height, divided by 3.
A cone holds one third of the cylinder that shares its 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 base is a circle whose area is pi times its radius squared, so the base is measured in one step.
The cylinder's base area
Compute the cylinder's base area.
Grade 7 circle area: π r², and doubling r multiplies area by 4.
7.G.B.4Identify SubproblemsDivide volume by area
Dividing cancels the pi.
Grade 6 equations: π cancels, and units cm³ ÷ cm² = cm — the answer is a length.
6.EE.B.7Analyze The UnitsSimplify
It tidies into a clean value.
Grade 5 fractions: 864 ÷ 288 = 3 and 576 ÷ 288 = 2, so the quotient simplifies to 3/2.
5.NF.B.3Analyze The UnitsMatch the choice
The height is 1.5 centimetres.
Grade 5 decimals: 1.5 is exactly choice (A).
5.NBT.A.3Eliminate PossibilitiesThis AMC 12 problem only needs Grade 8 cone-volume formula 1/3π r² h you already know — the cone holds 864π cm³ of water, the wider cylinder has base 576π cm², so the water height is 864 / 576 = 1.5 cm, choice (A).