Multiple choice

A solid metallic right circular cylinder and a solid metallic right circular cone are given. The cylinder and cone both have same height $h$ and same base radii $r$. The two solids are melted together and recast into a solid cylinder of radius $\displaystyle \frac{1}{2} r $. Then the height of the cylinder is $\displaystyle \frac{16}{3} h$.

  1. True

  2. False
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A Correct answer
Explanation

Volume of cylinder = pi*r^2*h. Volume of cone = (1/3)pi*r^2*h. Total volume = (4/3)*pi*r^2*h. New cylinder has radius r/2, so volume = pi(r/2)^2*H = pi*(r^2/4)*H. Equating volumes: (4/3)*pi*r^2*h = (1/4)*pi*r^2*H. H = (16/3)*h.

AI explanation

The sum of the volumes of the original cylinder and cone is (pi * r^2 * h) + (1/3 * pi * r^2 * h), which equals 4/3 * pi * r^2 * h. Equating this to the volume of the new cylinder with radius r/2 gives 4/3 * pi * r^2 * h = pi * (r/2)^2 * H. Solving for H yields H = 16/3 * h, making the statement true.

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