Maths Olympiad Prep

Library / /43 of 80

Geometry Difficulty 5.0 AIME Prove it North Macedonia

In a square shaped area on the ground with side length 12 m12\text{ m} a cylindrical hole is dug with diameter 8 m8\text{ m}. The dug soil is equally allotted on the square shaped area, outside of the hole, and then is pressed as much as was before the digging. How deep should be the dug in order the hole to be deep?

Solution

The area of the land where the dug soil is allotted is (122−42π) m2=(144−16π) m2(12^2 - 4^2\pi)\text{ m}^2 = (144 - 16\pi)\text{ m}^2. Let xx be the height of the hole before the dug soil is allotted. Then the thickness of the layer of allotted and then pressed soil is (3−x) m(3-x)\text{ m}. Because the dug soil is pressed as much as it was before the digging it, the volume is the same, i.e.

Figure 1

42π⋅x=(144−16π)⋅(3−x)4^2\pi \cdot x = (144 - 16\pi) \cdot (3 - x).

If we solve the last equation we get that x=(3−π3) mx = \left(3 - \frac{\pi}{3}\right)\text{ m}. Because π≈3.14\pi \approx 3.14 we get that x≈1.95 mx \approx 1.95\text{ m} should be the dug.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.