Implicit differentiation and related rates

[Panphobia]

Homework Statement

The spherical head of a snowperson is melting under the HOT sun at the rate of -160 cc/h (cubic centimetres per hour.) Find the rate at which the radius is changing when the radius r=16. Use cm/h for the units.
(The volume of a sphere is given by V= 4π⋅r^3/3.)

I have missed the past few calculus lectures and I am afraid I am falling behind, how would I start this kind of question? I know that the volume is changing at a rate of V - 160t where t is the number of hours...but I don't know how that helps at all.

 

[Tanya Sharma]

V = (4/3)πr³

Differentiate both sides with respect to ‘t’ .What do you get ?

 

[Panphobia]

dV/dt = 4πr^2*dr/dt

 

[Tanya Sharma]

dV/dt = 4πr^2*dr/dt

Excellent...

Now,dV/dt and 'r' is given to you .Just calculate dr/dt .

 

[Panphobia]

Oh my (facepalm) thank you so much for the help!

 

[Tanya Sharma]

:thumbs:

You are welcome

 

[Panphobia]

If I was to do the same thing but instead I was given a volume and was looking for the rate of change of volume given the rate of change of radius, would I just isolate for r then take the derivative?

 

[Tanya Sharma]

If I was to do the same thing but instead I was given a volume and was looking for the rate of change of volume given the rate of change of radius, would I just isolate for r then take the derivative?

That would be quite tedious .

Instead, from the given volume just find out the radius using the relation V =(4/3)πr³ .Then approach in the similar manner .