Question

Water is dripping out from a conical funnel, at the uniform rate of 2 cc/sec through a tiny hole at the vertex of the funnel. When the slant height of water is 5 cm, find the rate of decrease of the slant height of the water.

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Solution

Let V be volume of the curve whose semi-vertical angle is α, radius = r, height = h and slant height = l.

therefore space space space space space space straight V space equals space 1 third πr squared straight h Now space in space straight r. straight t. space angle straight d space increment OMP comma space space space space space space space space space space space space space space straight r over straight l space equals space sin space straight alpha comma space space space space straight h over straight l space equals space cos space straight alpha therefore space space space space space straight r space equals space straight l space sin space straight alpha comma space space space space straight h space equals space straight l space cos space straight alpha therefore space space space space space space straight V space equals space 1 third straight pi left parenthesis straight l squared space sin squared straight alpha right parenthesis thin space left parenthesis straight l space cos space straight alpha right parenthesis space equals space straight pi over 3 straight l cubed space sin squared straight alpha space cos space straight alpha therefore space space space space space space dV over dt space equals space straight pi over 3 space sin squared space straight alpha space cos space straight alpha. space open parentheses 3 straight l squared space dl over dt close parentheses rightwards double arrow space space space space dV over dt space equals space πl squared sin squared straight alpha space cos space straight alpha. space space dl over dt From space the space given space condition comma

dV over dt space equals space minus 2

therefore space space space space space space space πl squared space sin squared straight alpha space cos space straight alpha. space dl over dt space equals space minus 2 therefore space space space space space space space dl over dt space equals space minus fraction numerator 2 over denominator πl squared space sin squared straight alpha space cos space straight alpha end fraction When space straight l space equals space 4 space cm comma space we space have space space space space space space space dl over dt space equals space minus fraction numerator 2 over denominator 16 straight pi space sin squared space straight alpha space cos space straight alpha end fraction space equals space minus fraction numerator 1 over denominator 8 straight pi space sin squared space straight alpha space cos space straight alpha end fraction therefore space space space rate space of space decrease space of space slant space height space space equals space minus fraction numerator 1 over denominator 8 straight pi space sin squared space straight alpha space cos space straight alpha end fraction straight m divided by sec.

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Application Of Derivatives

Water is dripping out from a conical funnel, at the uniform rate of 2 cc/sec through a tiny hole at the vertex of the funnel. When the slant height of water is 5 cm, find the rate of decrease of the slant height of the water.

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