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Maths olympiad question · P6★★★★ Expert

Solids: olympiad question 11

An expert question on solids for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 first rounds and finals: painted small cubes, water levels and the surface area of stacks.

The question

A glass tank is 20 cm long, 20 cm wide and 35 cm high inside, with water 13 cm deep.

An iron block is 16 cm by 15 cm by 5 cm.

It is put in three different ways, lying flat on the bottom (one way at a time; it is taken out between, so the water is back to 13 cm).

The depths with the block in are x, y and z cm; no water spills.

What is x + y + z, the three water depths added, in cm?

Show the hints (3)
  1. Hint 1: With the block in, does the water cover it? Take each way separately.
  2. Hint 2: First suppose the block sticks out: the water fills only round it, over 400 − the block's footprint, depth = water ÷ that area. If that comes out higher than the block, the block is under after all.
  3. Hint 3: The water is 400 × 13 = 5200 cm³ and the block 1200 cm³. When the block is under, the depth = (5200 + 1200) ÷ 400.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Water = 20 × 20 × 13 = 5200 cm³; floor 400 cm²; block 16 × 15 × 5 = 1200 cm³.
  2. 2.Way 1, the 16 × 15 face down (5 cm tall): if it stuck out, the depth would be 5200 ÷ (400 − 240) = 5200 ÷ 160, more than 5 cm — so the block is under. Depth = (5200 + 1200) ÷ 400 = 16 cm.
  3. 3.Way 2, the 16 × 5 face down (15 cm tall): if it stuck out, the depth would be 5200 ÷ (400 − 80) = 5200 ÷ 320, more than 15 cm — so the block is under. Depth = (5200 + 1200) ÷ 400 = 16 cm.
  4. 4.Way 3, the 15 × 5 face down (16 cm tall): if it sticks out, the depth is 5200 ÷ (400 − 75) = 16 cm, no more than 16 cm — so it does stick out.
  5. 5.x + y + z = 16 + 16 + 16 = 48.
  6. ⚠️Common mistake: using one formula for all three. Some ways leave the block sticking out (water only beside it), others cover it (water and block fill the floor together) — decide first, then work it out.

Answer: 48 cm

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