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BAISHALI Rath

Mar 9, 2015

# A sphere and a cube have the same surface area . show that the ratio of the volumes of the sphere to that of the cube is âˆš6 : âˆšÏ€

A sphere and a cube have the same surface area . show that the ratio of the volumes of the sphere to that of the cube is âˆš6 : âˆšÏ€.

Raghunath Reddy

Let r and a be the radius of the sphere and edge of the cube respectively. Given, Surface area of sphere = Surface area of cube 4Ï€r2 = 6a2 (r/a)2 = 3 / 2Ï€ r / a = âˆš(3/2Ï€) Volume of sphere / Volume of cube = (4/3)Ï€r3 / a3 = (4Ï€/3)(r/a)3 = (4Ï€/3)(âˆš(3/2Ï€))3 = (4Ï€/3)(3/2Ï€)(âˆš(3/2Ï€)) = 2âˆš(3/2Ï€) = âˆš(4x3/2Ï€) = âˆš(6/Ï€) Thus, Volume of sphere : Volume of cube = âˆš6 : âˆšÏ€.

STAFF-95393

$\fn_cm \\Solution:\\ Given:\\ Surface\ area\ of\ a\ sphere= Surface\ area\ of\ a\ cube\\\\ \Rightarrow 4\pi r^2=6a^2\\\\ \Rightarrow \frac{r^2}{a^2}=\frac{6}{4\pi}\ \ \ \rightarrow (1)\\\\ \Rightarrow \frac{r}{a}=\sqrt{\frac{6}{\pi}}\\ \\\Rightarrow \frac{r}{a}=\frac{1}{2}\sqrt{\frac{6}{\pi}}\ \ \ \rightarrow (2)\\ \\Now\\ The\ ratio\ of\ volumes\ of\ a\ shere\ and\ a\ cube\\\\ =\frac{\frac{4}{3}\pi r^3}{a^3}\\\\\\ =\frac{4\pi r}{3a}\frac{r^2}{a^2}\\\\\\ From\ (1)\ and\ (2)\\\\ =\frac{4\pi }{3}\times \frac{1}{2}\sqrt{\frac{6}{\pi}}\times\frac{6}{4\pi}\\\\\\ =\sqrt{\frac{6}\pi}=\frac{\sqrt6}{\sqrt\pi}$
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