AMC 10 · 2014 · #19
Grade 8 geometry-3dFour cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XY contained in the cube with edge length 3?
Pick an answer.
AMC 10 2014 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.
Toolkit + CCSS Solution
Understand
Restated: Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked in a staircase tower that shares one vertical corner post. $X$ is the top corner of the smallest cube (top of the post) and $Y$ is the far bottom corner of the largest cube. Find how much of the straight segment $\overline{XY}$ lies inside the cube whose edge is $3$.
Givens: Cubes have edge lengths $1$, $2$, $3$, $4$, stacked smallest on top and largest on the bottom; All four cubes are aligned so they share one common vertical corner edge (the post); $X$ is the top corner of the post; $Y$ is the bottom corner diagonally across the base of the edge-$4$ cube; Answer choices: (A) $\dfrac{3\sqrt{33}}{5}$, (B) $2\sqrt3$, (C) $\dfrac{2\sqrt{33}}{3}$, (D) $4$, (E) $3\sqrt2$
Unknowns: The length of the part of $\overline{XY}$ that passes through the edge-$3$ cube
Understand
Restated: Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked in a staircase tower that shares one vertical corner post. $X$ is the top corner of the smallest cube (top of the post) and $Y$ is the far bottom corner of the largest cube. Find how much of the straight segment $\overline{XY}$ lies inside the cube whose edge is $3$.
Givens: Cubes have edge lengths $1$, $2$, $3$, $4$, stacked smallest on top and largest on the bottom; All four cubes are aligned so they share one common vertical corner edge (the post); $X$ is the top corner of the post; $Y$ is the bottom corner diagonally across the base of the edge-$4$ cube; Answer choices: (A) $\dfrac{3\sqrt{33}}{5}$, (B) $2\sqrt3$, (C) $\dfrac{2\sqrt{33}}{3}$, (D) $4$, (E) $3\sqrt2$
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #1 Draw a Diagram, #4 Introduce a Variable, #7 Identify Subproblems
The picture is a 3D staircase, so Tool #17 (Visualize Spatial Relationships) is the anchor: we must picture where $X$, $Y$, and the middle cube sit in space before any numbers help. Tool #1 (Draw a Diagram) turns that mental picture into $xyz$-coordinates so lengths become computable. Tool #7 (Identify Subproblems) splits the task into two clean pieces — first the full length of $\overline{XY}$, then the slice inside the edge-$3$ cube. Tool #4 (Introduce a Variable) lets us slide a point along $\overline{XY}$ to see exactly which part of it is at the height of the edge-$3$ cube.
Execute — Answer: A
6.NS.C.8 Step 1 Set coordinates and place X and Y
- Put the shared vertical post along the $z$-axis, with each cube reaching out in the $x$ and $y$ directions.
- Stacking largest at the bottom, the edge-$4$ cube fills heights $0$ to $4$, the edge-$3$ cube fills $4$ to $7$, the edge-$2$ cube fills $7$ to $9$, and the edge-$1$ cube fills $9$ to $10$.
- So $X$, the top of the post, is at $(0,0,10)$, and $Y$, the opposite bottom corner of the edge-$4$ cube, is at $(4,4,0)$.
💡 Anchoring the tower to the coordinate axes turns "which corner" into exact numbers you can measure.
8.G.B.7 Step 2 Find the full length of XY
- From $X$ to $Y$ the point moves $4$ in the $x$-direction, $4$ in the $y$-direction, and drops $10$ in height.
- The horizontal move alone is the base diagonal of the edge-$4$ cube, $\sqrt{4^2+4^2}=4\sqrt2$.
- Combining the horizontal move with the $10$ drop using the Pythagorean theorem in three dimensions gives the whole length: $XY=\sqrt{(4\sqrt2)^2+10^2}=\sqrt{32+100}=\sqrt{132}=2\sqrt{33}$.
💡 A straight segment in space is the hypotenuse of a right-angle box, so squaring the three moves and adding them gives its length.
7.RP.A.2 Step 3 Take the slice at the edge-3 cube's height
- The segment falls steadily, so its length grows in step with how far it drops.
- The edge-$3$ cube covers heights $4$ to $7$, a vertical band of $3$ out of the total drop of $10$.
- That band is the fraction $\tfrac{3}{10}$ of the segment, so the piece inside is $\tfrac{3}{10}\cdot 2\sqrt{33}=\tfrac{3\sqrt{33}}{5}$.
- Check the footprint: as the point crosses this band it moves from $(1.2,1.2)$ to $(2.4,2.4)$ in $x$ and $y$, both under $3$, so it truly stays inside the edge-$3$ cube the whole way.
- The length is $\dfrac{3\sqrt{33}}{5}$, choice (A).
💡 Because the line's length and its height loss stay in fixed proportion, a slice worth $3$ of the $10$ height is worth $\tfrac{3}{10}$ of the length.
6.NS.C.8 Put the shared vertical post along the $z$-axis, with each cube reaching out in 8.G.B.7 From $X$ to $Y$ the point moves $4$ in the $x$-direction, $4$ in the $y$-directi 7.RP.A.2 The segment falls steadily, so its length grows in step with how far it drops. T Review
Reasonableness: The full segment is $2\sqrt{33}\approx 11.5$, and the edge-$3$ cube is $3$ of the tower's $10$ units tall, so its slice should be a bit under a third of $11.5$, roughly $3.4$. The answer $\tfrac{3\sqrt{33}}{5}\approx 3.45$ matches. It is also correctly larger than the edge-$1$ or edge-$2$ slices would be and smaller than the edge-$4$ slice, since a taller cube captures a longer piece of the same slanted line. The footprint check ($x,y$ reaching only $2.4<3$) confirms the segment never leaks out a side face, so the whole height band counts.
Alternative: Instead of a fraction, parametrize the segment as $(4t,4t,10-10t)$ for $t$ from $0$ to $1$. The edge-$3$ cube's heights $4\le 10-10t\le 7$ give $0.3\le t\le 0.6$, a $t$-interval of length $0.3$. Since the full segment corresponds to $t$ running over length $1$ and measures $2\sqrt{33}$, the slice measures $0.3\cdot 2\sqrt{33}=\tfrac{3\sqrt{33}}{5}$, the same answer.
CCSS standards used (min grade 8)
6.NS.C.8Solve real-world and mathematical problems by graphing points in the coordinate plane (Assigning $xyz$-coordinates to the tower so that $X=(0,0,10)$ and $Y=(4,4,0)$ become exact points to measure between.)8.G.B.7Apply the Pythagorean theorem to determine unknown side lengths in right triangles in two and three dimensions (Combining the horizontal move $4\sqrt2$ with the vertical drop $10$ to get the full length $XY=2\sqrt{33}$.)7.RP.A.2Recognize and represent proportional relationships between quantities (Using that segment length is proportional to height dropped, so the height-$3$ band takes $\tfrac{3}{10}$ of the length.)
⭐ A straight slanted line loses length in step with height, so the cube that is $3$ tall out of $10$ swallows exactly $\tfrac{3}{10}$ of the segment: $\tfrac{3}{10}\cdot 2\sqrt{33}=\tfrac{3\sqrt{33}}{5}$.
⭐ A straight slanted line loses length in step with height, so the cube that is $3$ tall out of $10$ swallows exactly $\tfrac{3}{10}$ of the segment: $\tfrac{3}{10}\cdot 2\sqrt{33}=\tfrac{3\sqrt{33}}{5}$.
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