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Question

If water is poured into an inverted hollow cone whose semi-vertical angle is 30 , and its depth (measured along axis) increase at the rate of 1 cm per sec, find the rate at which the volume of water is increasing when the depth is 24 cm.

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Solution


Let the height of cone =h
Radius =r
θ=30°
According to question, dhdt=1cm/sec
tanθ=rh13=rhr=h3(1)Now,dvdt=ddt(13πr2h)=13πddt(r2h)
Using (1)
dvdt=13πddt(h23h)=13π13ddt(h3)dvdt=19π3h2dhdt=π3h2dhdtWhen h=24cmdvdt=π32424(1)=192πcm2/sec


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