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Maths-

General

Easy

Question

If the semi-vertical angle of a cone is $45 degree$ , then the rate of change of volume of the cone is

Curved surface area times the rate of change of $r$
Base area times the rate of change of $l$
Base area times the rate of change of $r$
None of these

The correct answer is: Base area times the rate of change of $l$

Let $r$ be the radius of the base, $h$ the height and $l$ slant height of the cone with semi-vertical angle $45 degree$ . Then,
$r equals h$ and $l to the power of 2 end exponent equals r to the power of 2 end exponent plus h to the power of 2 end exponent equals 2 r to the power of 2 end exponent$

Let $V$ be the volume. Then,

$V equals fraction numerator 1 over denominator 3 end fraction pi r to the power of 2 end exponent h equals fraction numerator 1 over denominator 3 end fraction pi r to the power of 3 end exponent$

$rightwards double arrow fraction numerator d v over denominator d t end fraction equals pi r to the power of 2 end exponent fraction numerator d r over denominator d t end fraction equals A r e a blank o f blank t h e blank b a s e cross times fraction numerator d r over denominator d t end fraction$

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