Math CentralQuandaries & Queries


Question from Lyudmyla, a student:

How fast is the volume of a cone increasing when the radius of its base is 2 cm and growing at a rate of 0.4 cm/s, and its height is 5 cm and growing at a rate of 0.1 cm/s?


The volume of a cone is given by

V = 1/3 π r2 h

where r is the radius and h is the height. The radius and the height of your cone are changing so r, h and V are functions of t. Differentiate both sides of the equation with respect to t. You need to use the product rule so you get

dV/dt = 1/3 π [d(r2)/dt] h + 1/3 π r2 dh/dt

which is

dV/dt = 1/3 π 2r [dr/dt] h + 1/3 π r2 dh/dt


dV/dt = 2/3 π rh dr/dt + 1/3 π r2 dh/dt

Substitute the values you know and solve for dV/dy.


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