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Maths-

General

Easy

Question

A conical vessel of height 10ft and semivertical angle $30 to the power of ring operator end exponent$ is full of water. If empties in such away that the height at the water in the vessel is decreasing at a constant rate of 1ft/min. Then the rate at which the volume of water in the vessel is decreasing when its height is 6ft. is

$π$
$12 π$
$\frac{π}{2}$
$3 π$

hint Hint:

Rate of change of volume with respect to time ( $fraction numerator d v over denominator d t end fraction$ ) = $1 third straight pi fraction numerator d over denominator d t end fraction left parenthesis r squared straight h right parenthesis$

The correct answer is: $12 pi$

Given :
$theta space equals space 30 degree h space equals space 10 f t fraction numerator d h over denominator d t end fraction space equals space 1 f t divided by m i n r a t e space o f space c h a n g e space left parenthesis h right parenthesis space equals space 6 f t$
We know that, tan $theta$ = $fraction numerator o p p o s i t e space s i d e over denominator a d j a c e n t space s i d e end fraction$
tan $theta$ = $r over h$
$tan 30 degree space equals space r over h tan 30 degree space equals space fraction numerator 1 over denominator square root of 3 end fraction r space equals space fraction numerator h over denominator square root of 3 end fraction$
Volume of conical vessel (V) = $1 third πr squared straight h$
V = $1 third straight pi straight h squared over 3 straight h space equals space 1 over 9 πh cubed$
Differentiating the above wrt to time
$fraction numerator d v over denominator d t end fraction space equals space 1 over 9 straight pi straight d over dt left parenthesis straight h cubed right parenthesis straight d over dt open parentheses straight x to the power of straight n close parentheses space equals space straight n space. space straight x to the power of straight n minus 1 end exponent space cross times space dx$
$fraction numerator d v over denominator d t end fraction space equals space straight pi over 9 cross times 3 h squared space cross times fraction numerator d h over denominator d t end fraction S u b s t i t u t i n g space t h e space v a l u e s space o f space h space a n d space fraction numerator d h over denominator d t end fraction fraction numerator d v over denominator d t end fraction space equals space space straight pi over 9 cross times space 3 space cross times left parenthesis 6 right parenthesis squared cross times 1 fraction numerator d v over denominator d t end fraction space equals space 12 straight pi$
Thus, the rate at which the volume of water in the vessel is decreasing when its height is 6ft. is $12 straight pi$

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