AMC 10 · 2010 · #12
Grade 8 geometry-3dPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #8 (Analyze the Units): the two given amounts are volumes (liters), but the answer is a length (meters). Volume measures space in three directions, so it grows like length cubed. That single fact links the volume ratio to the length ratio and is the whole key. Tool #7 (Subproblems): the work splits cleanly into three small jobs — find the volume ratio, take its cube root to get the length ratio, then shrink the height. Tool #3 (Eliminate): a quick reverse check that 100³ really is a million confirms one choice and rules out the decoys built from π.
Compare the two volumes
Divide the real volume by the model volume: the real sphere holds 1,000,000 times as much.
The volume ratio just asks how many tiny models it would take to fill the real sphere.
6.RP.A.1Identify SubproblemsVolume grows like length cubed
A liter measures space in three directions at once, so scaling every length by k scales the volume by k³.
Stretching a box by k in each of its three directions multiplies the space inside by k three times.
Stretching a solid by the same factor in all three directions multiplies the space inside three times over.
▸ Why?
Volume is length times width times height, so each direction contributes its own factor.
▸ Why?
Area already grows with the square of the scaling, and the third direction adds one more factor.
Undo the cube
Cube-root the million: 100 × 100 × 100 = 1,000,000, so every real length is 100 times the model's.
Volume shrinks fast, so a huge million-to-one volume gap comes from only a hundred-to-one gap in length.
8.EE.A.2Analyze The UnitsShrink the height
Height is a length, so it shrinks by that same 100: 40 ÷ 100 = 0.4 meter, which is choice (C).
Once you know lengths shrink by 100, every length — including the height — divides by 100.
5.NBT.B.7Identify SubproblemsCheck and rule out the rest
Scale back up to check: 0.1 × 100³ = 100,000 liters. The formula's π sits on both towers and cancels, so (B) and (D) die.
A correct scale must rebuild the original volume when you cube it back, and shared factors like π never survive a ratio.
6.RP.A.3Eliminate PossibilitiesVolume grows by length cubed, so a million-times-bigger volume means only a hundred-times-bigger length: 40 ÷ 100 = 0.4 m.
- Compare the two volumes
- Volume grows like length cubed
- Undo the cube
- Shrink the height
- Check and rule out the rest