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Sand is pouring from a pipe at the rate of 12 {c}{m}^{{3}}/{s}

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Sand is pouring from a pipe at the rate of 12 $c m^{3} / s$ . The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm.

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Sand is pouring from a pipe at the rate of 12

c m^{3} / s

. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm.

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Text solutionVerified

Solution:

h = \frac{1}{6} r

r = 6 h

\frac{d v}{d t} = 12 \frac{c m ^{3}}{s}

\frac{d h}{d t} = ?

v = (\frac{1}{3} π r^{2} h)

v = (h)

= \frac{1}{3} π (6 h)^{2} h

= \frac{36 π h ^{3}}{3}

= 12 π h^{3}

\frac{d v}{d t} = 12 \frac{c m ^{3}}{sec}

= d (12 π h^{3}) = 12

= 12 π \frac{d h ^{3}}{d t} = 12

\frac{d h}{d t} = \frac{1}{π} (\frac{1}{\frac{d h ^{3}}{d h}})

= \frac{1}{π} (\frac{1}{3 h ^{2}})

\frac{d h}{d t} (h = 4 c m) = \frac{1}{π} (\frac{1}{3 ( 4 ) ^{2}})

= \frac{1}{48 π} \frac{c m}{s}

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Question Text	Sand is pouring from a pipe at the rate of 12 $c m^{3} / s$ . The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm.
Answer Type	Text solution:1
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Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm.

Text solutionVerified

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Sand is pouring from a pipe at the rate of 12 $c m^{3} / s$ . The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm.