AMC 10 · 2025 · #15

Grade 7 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid
volume-pyramidsimilar-figuresratio-proportion identify-subproblems ↑ Prerequisites: ratio-proportion
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A tank has a 1×1 square bottom, a 3×3 open square top, and four matching slanted sides. Water fills it at a steady rate. Reaching the midline of the slanted sides (the halfway height) takes 35 minutes. Find how many more minutes are needed to finish filling the tank.

Pick an answer.

(A)
70
(B)
85
(C)
90
(D)
95
(E)
105
How to solve
Strategy Solve an Easier Related Problem

The tank is a frustum (a pyramid with its tip sliced off), and there is no clean K-8 formula for its volume. Tool #9 (Solve an Easier Related Problem) fixes this: extend the four slanted sides downward until they meet at a point, turning the awkward frustum into a full square pyramid. Then the shape is made of three nested pyramids whose square bases have sides 1, 2, and 3. Tool #17 (Visualize Spatial Relationships) sees that these three pyramids are scaled copies, so their volumes compare as cubes. Tool #1 (Draw a Diagram) tracks the growing square slice, and Tool #8 (Analyze the Units) turns the volume ratio into a time by using the constant fill rate.

1STEP 1

Complete the tip into a full pyramid

Complete it into a full pyramid.

tank = (full pyramid) - (cut-off tip)
2STEP 2

Slices are squares of side 1, 2, 3

The cross-sections have sides 1, 2, 3.

side at midline = (1+3)/2 = 2
3STEP 3

Volumes scale as the cube: 1, 8, 27

Volumes go as 1 to 8 to 27.

1³ : 2³ : 3³ = 1 : 8 : 27
4STEP 4

Filled part is 7, remaining part is 19

Filled is 7, remaining is 19.

8 - 1 = 7 27 - 8 = 19
5STEP 5

Convert volume to time

Five minutes per unit gives 95 minutes.

35/7 = 5 min/unit → 19 × 5 = 95
Answer
95
The upper half of the tank is much wider than the lower half (a 2-to-3 square band versus a 1-to-2 band), so it should hold far more water and take much longer than 35 minutes. The ratio 19:7 gives 35 × 19/7 = 95 minutes, comfortably more than 35, which fits. A quick cross-check with slice areas: the filled volume behaves like 2³ - 1³ = 7 and the remaining like 3³ - 2³ = 19, the same 7:19 split, confirming (D). Choices 70 and 85 are too small for how much wider the top is.
💡Key takeaway

Fill in a cut-off pyramid's missing tip, then compare scaled copies by cubing the scale — volume grows as the cube of the size, so a slightly wider top can hold a lot more water.

  • Complete the tip into a full pyramid
  • Slices are squares of side 1, 2, 3
  • Volumes scale as the cube: 1, 8, 27
  • Filled part is 7, remaining part is 19
  • Convert volume to time

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