Problem 15 · 2011 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
A marble of radius 15 is rolled into a cone-shaped hole. It fits in perfectly. From the side the cone looks like an equilateral triangle. How deep is the hole?
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Answer: C — 45
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Hint 1 of 2
From the side the cone is an equilateral triangle and the marble is its inscribed circle.
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Hint 2 of 2
For an equilateral triangle the inradius is one third of the height.
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Approach: use the equilateral triangle's inradius
- Side-on, the marble is the circle inscribed in an equilateral triangle, with radius 15.
- In an equilateral triangle the inradius equals one third of the height.
- So the height (the hole's depth) = 3 × 15 = 45.
- The hole is 45 deep, choice (C).
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