Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer United States

Problem:
On a spherical planet with diameter 10,000 km10,000~\mathrm{km}, powerful explosives are placed at the north and south poles. The explosives are designed to vaporize all matter within 5,000 km5,000~\mathrm{km} of ground zero and leave anything beyond 5,000 km5,000~\mathrm{km} untouched. After the explosives are set off, what is the new surface area of the planet, in square kilometers?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
100,000,000π100,000,000 \pi

The explosives have the same radius as the planet, so the surface area of the "cap" removed is the same as the new surface area revealed in the resulting "dimple." Thus the area is preserved by the explosion and remains π⋅(10,000)2\pi \cdot (10,000)^2.

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