How fast is the volume of a cylinder changing with respect to the radius when the radius is mm and the height is a constant 5mm?

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Answer 1 · 2017-12-11T14:41:55

The change will be proportional to the square of the radius.

Given two cylinders with the same height and radii r1 and r2 their volume will be:

#V_1=hpir_1^2# and #V_2=hpir_2^2#

The ratio of volumes between them will be:

#V_2/(V_1)=(hpir_2^2)/(hpir_1^2)=(r_2^2)/(r_1^2)#

This means that these two cylinders are correlated by the square of radius.

Answer 2 · 2017-12-11T16:00:16

#(dV)/(dr) = 10pir#

The general formula for the volume of a cylinder is #V = pir^2h#

We are told that the height is constant, so this cylinder has volume #V = 5pir^2#

The rate of change of volume with respect to radius is #(dV)/(dr)#

#d/(dr)(V) = d/(dr)(5pir^2) = 5pi d/(dr)(r^2) = 5pi(2r) = 10pir#

The volume is changing at a rate of

#10pir# #mm^3"(of Volume)"# / #mm"(of radius)"#