AMC 10 · 2017 · #11

Grade 8 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Sum of Primitive Solids
volume-cylindervolume-spherespatial-visualization spatial-visualizationidentify-subproblems ↑ Prerequisites: volume-cylinder
📏 Medium solution 💡 2 insights
Problem
Take a straight segment AB and color every point in space that lies within 3 units of it. That colored solid has volume 216π. How long is AB?

Pick an answer.

(A)
6
(B)
12
(C)
18
(D)
20
(E)
24

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

The whole problem hinges on seeing the shape, so Tool #17 (Visualize Spatial Relationships) comes first: the points within 3 of a straight segment form a tube (cylinder) wrapped around the middle, capped by a rounded dome (hemisphere) at each end. Once the picture is clear, Tool #7 (Identify Subproblems) splits the volume into pieces we already have formulas for — a cylinder plus the two end domes, which together make one full sphere of radius 3. Tool #4 (Introduce a Variable) names the unknown length L, turns the total volume into the equation 9π L + 36π = 216π, and we solve for L.

1STEP 1

Picture the solid

Points within 3 of the segment sweep a radius-3 cylinder, and each end adds a radius-3 hemisphere cap — a cylinder plus two caps.

solid = (cylinder, radius 3) + (two hemisphere caps, radius 3)
2STEP 2

Combine the two caps into a sphere

The two radius-3 hemisphere caps join into one whole radius-3 ball, whose sphere volume 43\frac{4}{3}·π·3³ is 36π.

2·1/2·4/3π r³ = 4/3π(3)³ = 36π
3STEP 3

Write the cylinder volume with a variable

Let L be AB's length, the cylinder's height; its volume is π·3²·L = 9πL, so the whole solid is 9πL + 36π.

V_cyl = π(3)² L = 9π L, V_total = 9π L + 36π
4STEP 4

Set the volume equal and solve

Set 9πL + 36π = 216π; dividing by π gives 9L + 36 = 216, so 9L = 180 and L = 20 — answer (D).

9π L + 36π = 216π → 9L + 36 = 216 → 9L = 180 → L = 20 → (D)
Answer
20
Check by plugging L=20 back in: the cylinder gives 9π(20)=180π, the two caps give 36π, and 180π+36π=216π, exactly the stated volume. The answer is also sensible: the caps alone account for only 36π of the 216π, so most of the solid is the long tube, which fits a length much bigger than the radius. Length 20 is choice (D).
💡Key takeaway

Everything within a fixed distance of a segment is a tube with a domed cap on each end — add the cylinder and the two caps (a whole sphere), then solve for the length.

  • Picture the solid
  • Combine the two caps into a sphere
  • Write the cylinder volume with a variable
  • Set the volume equal and solve

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