AMC 10 · 2010 · #7
Grade 8 geometry-3dPick an answer.
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
The capacities differ by a factor of a million.
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
Volume grows like length cubed.
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 each of its three directions multiplies the space inside three times over.
▸ Why?
Volume is base area repeated along a height, so both the base and the height carry the scale factor.
▸ Why?
The base area already picks up the factor twice, once for each of its two directions.
Undo the cube
Undoing the cube gives a length ratio of 100.
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
Shrinking the height gives 0.4 metres.
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
Scaling back up returns the real capacity, so 0.4 holds.
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