AMC 10 · 2004 · #25

Grade 8 geometry-3d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length

pythagorean-theoremtangent-circlescentroid-2-to-1 physical-representationidentify-subproblems ↑ Prerequisites: pythagorean-theorem

📏 Long solution 💡 2 insights

Problem

Three mutually tangent spheres of radius $1$ rest on a horizontal plane. A sphere of radius $2$ rests on them. What is the distance from the plane to the top of the larger sphere?

Pick an answer.

(A)

$\ 3 + \frac {\sqrt {30}}{2}$

(B)

$\ 3 + \frac {\sqrt {69}}{3}$

(C)

$\ 3 + \frac {\sqrt {123}}{4}$

(D)

$\ \frac {52}{9}$

(E)

$\ 3 + 2\sqrt2$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Three balls of radius 1 sit on a flat floor, each touching the other two. A bigger ball of radius 2 rests on top of all three. Find the height from the floor to the very top of the big ball.

Givens: Three spheres of radius 1 rest on a horizontal plane and are mutually tangent.; A sphere of radius 2 rests on top of the three small spheres.; Tangent spheres touch, so the distance between two sphere centers equals the sum of their radii.

Unknowns: The vertical distance from the plane to the highest point of the radius-2 sphere.

Understand

Plan

Primary tool: #17 Visualize Spatial Relationships

Secondary: #7 Identify Subproblems, #1 Draw a Diagram

The spheres themselves are hard to reason about, but their centers form a simple tetrahedron: an equilateral-triangle base of the three small centers with the big center as apex. Picturing that skeleton turns a 3D puzzle into two flat right triangles, each solved by the Pythagorean theorem. Splitting the total height into three stacked pieces (base-center height, the vertical rise to the apex, and the top radius) keeps every step small.

Execute — Answer: B

#17 Visualize Spatial Relationships 7.G.A.2 Step 1

Connect the four centers

Replace each sphere by its center.
The three small centers each sit 1 above the floor and are 2 apart, so they form an equilateral triangle of side 2 lying in a horizontal plane at height 1.
The big center is 3 away from every small center, and by symmetry it floats directly above the triangle's center.

$$d_{\text{small-small}} = 1 + 1 = 2, \qquad d_{\text{small-big}} = 1 + 2 = 3$$

💡 Two balls touch exactly when their centers are as far apart as their radii add up to.

#7 Identify Subproblems 8.G.B.7 Step 2

Radius of the base triangle

Find how far each small center is from the centroid directly below the big center.
In an equilateral triangle of side 2, the median has length $\sqrt3$, and the centroid lies two-thirds of the way down each median from a vertex.
So the distance from a vertex to the centroid (the circumradius) is two-thirds of $\sqrt3$.

$$R = \tfrac{2}{3}\sqrt3 = \frac{2}{\sqrt3} = \frac{2\sqrt3}{3}, \qquad R^2 = \frac{4}{3}$$

💡 The center of an equilateral triangle is two-thirds of the way down each median from the corner.

#7 Identify Subproblems 8.G.B.7 Step 3

Vertical rise to the top center

Look at the right triangle with corners at the big center, the centroid, and one small center.
The slanted side (center to center) is 3, the horizontal side is R, and the vertical side is the rise $h$ we want.
Apply the Pythagorean theorem to get $h$.

$$h = \sqrt{3^2 - R^2} = \sqrt{9 - \tfrac{4}{3}} = \sqrt{\tfrac{23}{3}}$$

💡 The center-to-center segment is the hypotenuse; its horizontal and vertical shadows are the two legs.

#7 Identify Subproblems 8.EE.A.2 Step 4

Stack the three heights

The top of the big sphere is the base-center height (1) plus the vertical rise to the big center ($h$) plus the big sphere's own radius (2).
Rationalize $h$ and add.
The total matches answer (B).

$$h = \sqrt{\tfrac{23}{3}} = \frac{\sqrt{23}}{\sqrt3} = \frac{\sqrt{69}}{3}, \qquad 1 + \frac{\sqrt{69}}{3} + 2 = 3 + \frac{\sqrt{69}}{3}$$

💡 Total height is just three stacked pieces: floor-to-center, center-to-center rise, and center-to-top radius.

[1] #17 7.G.A.2 Replace each sphere by its center. The three small centers each sit 1 above the

[2] #7 8.G.B.7 Find how far each small center is from the centroid directly below the big cente

[3] #7 8.G.B.7 Look at the right triangle with corners at the big center, the centroid, and one

[4] #7 8.EE.A.2 The top of the big sphere is the base-center height (1) plus the vertical rise t

Review

Reasonableness: Numerically $3 + \tfrac{\sqrt{69}}{3} \approx 3 + 2.77 = 5.77$. That is above 3 (the big center's own radius plus the base height already give more than 3) and below 6 (the vertical rise is under 3), so it lands in a sensible range. Notice every answer choice sits between about 5.74 and 5.83, so only the exact computation $h^2 = 23/3$ pins it to (B) rather than a near neighbor.

Alternative: Place coordinates: put the three small centers at height 1 with one at (0,0,1) and the centroid under the big center. Set the big center at (x_c, y_c, 1+h) where (x_c,y_c) is the centroid, and write the equation (distance from big center to a small center) = 3. Solving the single distance equation gives h = \sqrt{23/3} again, then add 1 and 2 for the same total.

CCSS standards used (min grade 8)

7.G.A.2 Draw geometric shapes with given conditions (Placing the four sphere centers from the tangency conditions (centers 2 apart and 3 apart).)
8.G.B.7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles (Finding the base triangle's circumradius and the vertical rise from the base plane to the big center.)
8.EE.A.2 Use square root and cube root symbols to represent solutions (Rationalizing $\sqrt{23/3}$ to $\sqrt{69}/3$ and assembling the final height.)

⭐ Swap the spheres for their centers, and a scary 3D stack becomes two flat right triangles you solve with the Pythagorean theorem.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Connect the four centers

Radius of the base triangle

Vertical rise to the top center

Stack the three heights

Review

CCSS standards used (min grade 8)

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