AMC 10 · 2005 · #22
Grade 9 geometry-3dPick an answer.
Naming the three dimensions turns both givens into symmetric equations in l, w, h. The sphere condition converts into the space diagonal, which needs l² + w² + h² — and that is precisely what appears when the sum l+w+h is squared. So the individual dimensions are never needed, only the two symmetric sums that were handed over.
Name the three dimensions
Both given facts become symmetric expressions in the three edges.
Two facts about the box become two symmetric equations in the same three unknowns.
9.A-CED.A.2Introduce A VariableThe long diagonal is the diameter
All eight corners are equidistant from the box's centre, so the long diagonal is the diameter.
All eight corners are equally far from the box's centre, so that point can only be the sphere's centre.
8.G.B.7Visualize Spatial RelationshipsPythagoras twice for the diagonal
Two right triangles stacked give that diagonal from the three edges.
A box diagonal is a flat diagonal with one more right-angle step added on.
8.G.B.7Draw A DiagramSquare the sum of the edges
Squaring the edge sum produces exactly what is needed: 400.
The two given totals are exactly the pieces that appear when a sum of three terms is squared.
Squaring the total of the three edges produces exactly the two quantities the problem already hands over.
▸ Why?
Expanding a sum multiplied by itself gives every square once and every cross term twice.
▸ Why?
The box diagonal squared is the three edge squares added together, so that sum is what the diagonal needs.
Halve the diagonal
Halving the diagonal gives 10, choice (B).
Once the diagonal is known, the radius is one division away.
8.EE.A.2Introduce A VariableSquaring l + w + h hands you the surface area for free, so the box's diagonal falls out without ever finding l, w, or h.
- Name the three dimensions
- The long diagonal is the diameter
- Pythagoras twice for the diagonal
- Square the sum of the edges
- Halve the diagonal