Sand is pouring from a pipe at the rate of 12 cm^3 / s. The falling sand forms a cone on the gr

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Sand is pouring from a pipe at the rate of 12 cm^3 / s. The...

Sand is pouring from a pipe at the rate of $12 \mathrm{~cm}^{3} / \mathrm{s}$. 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 $4 \mathrm{~cm}$ ?

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Key Concepts

Substitution Using Constraints

When a problem provides a relationship between variables (for instance, a fixed ratio between height and radius), it is possible to substitute one variable in terms of another. This reduces the number of variables in the problem, simplifying the process of differentiation and making it easier to solve for the desired rate.

Differentiation and the Chain Rule

In related rates problems, differentiating an equation with respect to time using the chain rule is a key technique. This method allows us to account for the interdependency between variables that are functions of time, thereby linking their rates of change.

Volume of a Cone

The volume of a cone is given by the formula V = (1/3)?r²h. This formula is essential when dealing with problems involving conical shapes, particularly when the volume is changing over time due to accumulation or removal of material, such as sand pouring to form a cone.

Related Rates

This concept involves finding the rate at which one quantity changes with respect to another, given that these quantities are related by an equation. In the context of calculus, related rates problems typically require differentiating an equation that connects the quantities with respect to time.

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