Multiple choice

A cylinder whose height is equal to $\displaystyle 2\frac{1}{4}$ times its diameter has the same volume as a sphere of radius 4 cm The radius of the base of the cylinder is

  1. $\displaystyle 3\frac{1}{2}$
  2. $3$ cm
  3. $\displaystyle 2\frac{1}{3}$
  4. $\displaystyle 2\frac{2}{3}$
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D Correct answer
Explanation

Volume of sphere = (4/3) * pi * r^3 = (4/3) * pi * 4^3 = 256/3 * pi. Cylinder volume = pi * R^2 * H. H = 2.25 * (2R) = 4.5R. pi * R^2 * 4.5R = 256/3 * pi. 4.5 * R^3 = 256/3. (9/2) * R^3 = 256/3. R^3 = 512/27. R = cube_root(512/27) = 8/3 = 2 and 2/3.

AI explanation

The height of the cylinder is $9/4$ times its diameter, meaning $h = (9/4) \times 2r = 9r/2$. The volume of this cylinder equals the volume of a sphere with radius $4$ cm, so $\pi r^2(9r/2) = (4/3)\pi (4)^3$. This simplifies to $9r^3/2 = 256/3$. Solving for $r^3$ gives $r^3 = 512/27$, so $r = 8/3$ cm. The radius of the base is $2\frac{2}{3}$ cm.

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