Multiple choice

A conical vessel of radius 12 cm and height 16 cm is filled with water. A sphere is lowered into water and its size is such that it touches the sides of the vessel and it is just immersed. What is the radius of the sphere?

  1. 5 cm

  2. 6 cm

  3. 6.5 cm

  4. 7 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using similar triangles in the cross-section, let r be the sphere radius. The distance from the cone vertex to the sphere center is 16 - r. By similarity, r / (16 - r) = 12 / sqrt(12^2 + 16^2) = 12 / 20 = 3/5. Solving 5r = 48 - 3r gives 8r = 48, so r = 6 cm.

AI explanation

Using the slant height formula for the cone, l = sqrt(12^2 + 16^2) = 20 cm, and the formula for the radius of an inscribed sphere in a cone, r = R*H / (R + l), we substitute the given values. This gives r = (12*16) / (12 + 20), which simplifies to 192 / 32. The result is a radius of 6 cm.

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