A sphere is melted to form a cylinder whose h | vedclass

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Question

(A) Let the radius of the sphere be $R$ and the radius of the cylinder be $r$.Given that the height of the cylinder $h = 4 \frac{1}{2} r = \frac{9}{2}

Vedclass · Accepted Answer

$3:2$

Vedclass · Answer

(A) Let the radius of the sphere be $R$ and the radius of the cylinder be $r$.Given that the height of the cylinder $h = 4 \frac{1}{2} r = \frac{9}{2} r$.Since the sphere is melted to form the cylinder,their volumes are equal.Volume of the sphere = $\frac{4}{3} \pi R^3$.Volume of the cylinder = $\pi r^2 h = \pi r^2 (\frac{9}{2} r) = \frac{9}{2} \pi r^3$.Equating the volumes: $\frac{4}{3} \pi R^3 = \frac{9}{2} \pi r^3$.Dividing both sides by $\pi$: $\frac{4}{3} R^3 = \frac{9}{2} r^3$.Rearranging for the ratio of radii: $\frac{R^3}{r^3} = \frac{9}{2} 	imes \frac{3}{4} = \frac{27}{8}$.Taking the cube root of both sides: $\frac{R}{r} = \sqrt[3]{\frac{27}{8}} = \frac{3}{2}$.Thus,the ratio of the radius of the sphere to the cylinder is $3:2$.

$A$ sphere is melted to form a cylinder whose height is $4 \frac{1}{2}$ times its radius. What is the ratio of the radius of the sphere to the radius of the cylinder?

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