Sunday, April 23, 2017

Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 80

Suppose that an air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is V(t) and its radius is R(t).

a.) What do the derivatives dVdr and dVdt represent?

dVdr represents how does the volume changes with respect to its radius. On the other hand, dVdt represents how quick the volume is changing with respect to time.


b.) Express dVdt in terms of drdt.

Recall that the volume of a sphere is V(r)=43πr3 and by using Chain Rule we have,


dVdt=dVdr(drdt)=4π3ddt(r3)=4π3ddr(r3)drdt=4π\cancel3(\cancel3r2)drdtdVdt=4πr2drdt

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