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Sand is pouring from a pipe at the rate ...

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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To solve the problem, we need to find the rate at which the height of the sand cone is increasing when the height is 4 cm. Let's go through the solution step by step. ### Step 1: Understand the relationship between the radius and height of the cone Given that the height \( h \) of the cone is always one-sixth of the radius \( r \) of the base, we can express this relationship mathematically as: \[ h = \frac{1}{6} r \quad \text{or} \quad r = 6h \] ...
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