A hard 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 cuboid tank is 30 cm long, 40 cm wide and 20 cm high inside, with water 9 cm deep.
An iron block 12 cm long, 20 cm wide and 10 cm high is stood upright on the bottom (shown); no water spills.
How deep is the water now, in cm?
Show the hints (3)
💡Hint 1: The water's volume doesn't change. Once the block is in, does it go right under, or stick out?
💡Hint 2: How much water would it take to reach 10 cm, the block's top? Compare with the water there is.
💡Hint 3: If it sticks out, the water only covers the floor beside the block.
Worked solution
Have a go first, then read the solution step by step.
1.The water's volume = 30 × 40 × 9 = 10800 cm³. Putting the block in doesn't add any.
2.First, can the water cover the block? To reach its top (10 cm), the space beside it needs (30 × 40 − 12 × 20) × 10 = 9600 cm³ of water. There is more than that, so the block goes right under.
3.With the block right under, the rise holds exactly the block's volume: rise = 12 × 20 × 10 ÷ (30 × 40) = 2 cm, depth = 9 + 2 = 11 cm.
⚠️Common mistake: seeing the block is taller than the water and assuming it sticks out, working 10800 ÷ (1200 − 240). The water rises once the block is in; compare the amounts first.
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