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Maths-

General

Easy

Question

The edge of a cube is equal to the radius of a sphere. If the edge and the radius increase at the same rate, then the ratio of the increases in surface areas of the cube and sphere is

$2 π : 3$
$3 : 2 π$
$6 : π$
None of these

The correct answer is: $3 blank colon 2 pi$

Let the length of an edge of the cube be $x$ units. Let $S subscript 1 end subscript$ and $S subscript 2 end subscript$ be the surface areas of the cube and sphere respectively. Then,
$S subscript 1 end subscript equals 6 x to the power of 2 end exponent$ and $equals S subscript 2 end subscript equals 4 pi x to the power of 2 end exponent$

$rightwards double arrow fraction numerator d S subscript 1 end subscript over denominator d t end fraction equals 12 x fraction numerator d x over denominator d t end fraction$ and $S subscript 2 end subscript equals 8 pi x fraction numerator d x over denominator d t end fraction rightwards double arrow fraction numerator fraction numerator d S subscript 1 end subscript over denominator d t end fraction over denominator fraction numerator d S subscript 2 end subscript over denominator d t end fraction end fraction equals fraction numerator 3 over denominator 2 pi end fraction$

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