AMC 10 · 2015 · #23
Grade 7 geometry-3dPick an answer.
The question asks "how many", so the goal is a complete, no-overlap count (Tool #2, Make a Systematic List). To make the list finite, first translate volume and surface area into the equation abc=2(ab+bc+ca) and divide to get frac1a+frac1b+frac1c=1/2 (Tool #4, Introduce a Variable). The Extreme Principle (Tool #14) then traps the smallest edge a in a tiny range {3,4,5,6}, and each value of a becomes a separate subproblem (Tool #7) solved by listing factor pairs.
Write the volume
Setting them equal gives one equation.
Filling the box with unit cubes makes a · b · c of them, so the volume is just the product.
6.G.A.2Introduce A VariableWrite the surface area and equate
Dividing makes it a sum of reciprocals.
Opposite faces of a box match, so you compute only three faces and double them.
Opposite faces of a box match, so you compute only three faces and double them.
▸ Why?
The surface is exactly its six faces put together, so their areas add to the total.
▸ Why?
Each face has exactly one twin opposite it, so counting three and doubling counts all six once.
Divide by abc
That bounds the smallest edge tightly.
Dividing turns a messy product equation into a tidy sum of fractions that is easy to bound.
7.EE.B.4Introduce A VariableBound the smallest edge a
Only four values are possible for it.
The smallest edge carries the biggest fraction, which squeezes a into just four values.
7.EE.B.4Extreme PrincipleCase a = 3
The first case factors into five boxes.
Rewriting as a product turns "find two fractions" into "list factor pairs", which is finite and easy to count.
4.OA.B.4Make A Systematic ListCase a = 4
The second gives three more.
Same factoring trick; a smaller leftover product means fewer factor pairs, so fewer triples.
4.OA.B.4Make A Systematic ListCase a = 5
The third gives one.
A non-unit fraction on the right leaves almost no room, so only one b survives the test.
7.EE.B.4Make A Systematic ListCase a = 6
The last gives one too.
At the top of a's range the edges are squeezed until all three are equal.
7.EE.B.4Make A Systematic ListAdd the cases
Adding gives 10, choice (B).
Separate cases (different smallest edge) just add up, with nothing double-counted.
2.NBT.B.5Make A Systematic ListTurn "volume equals surface area" into frac1a+frac1b+frac1c=1/2, squeeze the smallest edge into a few values, then count factor pairs in each case.
- Write the volume
- Write the surface area and equate
- Divide by abc
- Bound the smallest edge a
- Case a = 3
- Case a = 4
- Case a = 5
- Case a = 6
- Add the cases