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Mathematics

Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube.

Mensuration

ICSE

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Answer

Let edge of the cube = a units.

Then, Volume of the cube = a x a x a = a3

The sphere that exactly fits in the cube will have diameter a units.

Radius of sphere (r) = a2\dfrac{a}{2} units.

Vol. of cubeVol. of sphere=(side)343πr3=a343π×(a2)3=a3×3×234π×a3=3×84×227=3×8×74×22=2111=21:11.\dfrac{\text{Vol. of cube}}{\text{Vol. of sphere}} = \dfrac{\text{(side)}^3}{\dfrac{4}{3}πr^3} \\[1em] = \dfrac{a^3}{\dfrac{4}{3}π \times \Big(\dfrac{a}{2}\Big)^3} \\[1em] = \dfrac{a^3 \times 3 \times 2^3}{4π \times a^3} \\[1em] = \dfrac{3 \times 8}{4 \times \dfrac{22}{7}} \\[1em] = \dfrac{3 \times 8 \times 7}{4 \times 22} \\[1em] = \dfrac{21}{11} \\[1em] = 21 : 11.

Hence, ratio of volume of cube to volume of sphere = 21 : 11.

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