A sphere is melted to form a cylinder whose height is 2 ½ times its r

1.	A sphere is melted to form a cylinder whose height is 2 ½ times its radius. What is the ratio of the radius of sphere to cylinder?1). 1 : 42). 4 : 13). √5 : 24). \(\sqrt[3]{15}\;:2\)
Answer» Since the amount of material used for both the sphere and the material are the same, their volumes will be equal. Let the RADIUS of the sphere be r, and that of the cylinder be R. It is given that the height of the cylinder is 2 ½ times its radius, ⇒ H = 5/2 × R Volume of the sphere = $(\frac{4}{3}\pi {r^3})$ Volume of the cylinder = $(\pi {R^2}H = \pi {R^2} \times \frac{5}{2}R = \frac{5}{2}\pi {R^3})$ Since we KNOW that these volumes are the same, $(\begin{array}{L} \frac{4}{3}\pi {r^3}\; = \;\frac{5}{2}\pi {R^3}\\ \Rightarrow \;\frac{{{r^3}}}{{{R^3}}} = \frac{15}{8}\\ \Rightarrow \;r\;:\;R\; = \;\sqrt[3]{15}:2 \end{array})$

A sphere is melted to form a cylinder whose height is 2 ½ times its radius. What is the ratio of the radius of sphere to cylinder?1). 1 : 42). 4 : 13). √5 : 24). $\sqrt[3]{15}\;:2$

Answer»

Since the amount of material used for both the sphere and the material are the same, their volumes will be equal.

Let the RADIUS of the sphere be r, and that of the cylinder be R.

It is given that the height of the cylinder is 2 ½ times its radius,

⇒ H = 5/2 × R

Volume of the sphere = $(\frac{4}{3}\pi {r^3})$

Volume of the cylinder = $(\pi {R^2}H = \pi {R^2} \times \frac{5}{2}R = \frac{5}{2}\pi {R^3})$

Since we KNOW that these volumes are the same,

$(\begin{array}{L} \frac{4}{3}\pi {r^3}\; = \;\frac{5}{2}\pi {R^3}\\ \Rightarrow \;\frac{{{r^3}}}{{{R^3}}} = \frac{15}{8}\\ \Rightarrow \;r\;:\;R\; = \;\sqrt[3]{15}:2 \end{array})$

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