Multiple choice

A cylinder of radius 4.5 cm and height 12 cm just fits in another cylinder completely with their axes perpendicular. What is the radius (in cm) of the second cylinder?

  1. 5

  2. 6

  3. 15

  4. 7.5
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D Correct answer
Explanation

For a cylinder of radius r and height h to fit inside another cylinder of radius R with axes perpendicular, the cross-section of the inner cylinder must fit within the circular cross-section of the outer cylinder. This requires the diagonal of the inner cylinder's rectangle (2r by h) to be less than or equal to the diameter of the outer cylinder (2R). Thus, sqrt((2*4.5)^2 + 12^2) = 2R. sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15. So 2R = 15, R = 7.5.

AI explanation

For a cylinder of radius 4.5 cm and height 12 cm to fit completely inside another cylinder with perpendicular axes, the larger cylinder must have a diameter equal to the diagonal of the smaller cylinder's vertical cross-section. The diagonal is calculated using the Pythagorean theorem as sqrt(12^2 + (2 * 4.5)^2) = sqrt(144 + 81) = sqrt(225) = 15 cm. This diagonal is the diameter of the second cylinder, so its radius is 15 / 2 = 7.5 cm. The radius of the second cylinder is 7.5 cm.

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