Multiple choice

A right circular cone whose base radius is 4 is inscribed in a sphere of radius 5. What is the ratio of the volume of the cone to the volume of the sphere?

  1. 0.222 : 1

  2. 0.256 : 1

  3. 0.288 : 1

  4. 0.333 : 1

  5. 0.864 : 1
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B Correct answer
Explanation

For a cone of radius r=4 in a sphere of radius R=5, the height h is found via Pythagoras: (h-R)^2 + r^2 = R^2, so (h-5)^2 + 16 = 25, h-5 = 3, h=8. Volume of cone = (1/3)pi(r^2)h = (1/3)pi(16)(8) = 128pi/3. Volume of sphere = (4/3)pi(R^3) = (4/3)pi(125) = 500pi/3. Ratio = (128/3) / (500/3) = 128/500 = 0.256.

AI explanation

Using the Pythagorean theorem with the sphere's radius of 5, the cone's height is 5 + 3 = 8. The cone's volume is (1/3) (4^2)(8) = 128 /3, and the sphere's volume is (4/3) (5^3) = 500 /3. The ratio is (128 /3) / (500 /3) = 0.256, making the ratio 0.256 : 1.

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