This is a related rates problem, i've been having trouble with these for some reason. i think it's because the chain rule doesn't come naturally to me.

We are given that  (or  ).

a.
find the rate of change of volume when r = 12 m.
to relate the radius of a sphere to its volume, use the formula for the volume of a sphere.



differentiate both sides with respect to time.



we have r = 12m and dr/dt = 0.05 as information.



b.

find the rate of change of surface area when v = 36π m³.

the volume for surface area is 



differentiate both sides with respect to time.



we do not know the radius but we know the volume. we can find the radius knowing that volume is 36π m³.



so substituting information r = 3 and dr/dt = 0.05:



c.
this question asks us to express as a function of r. our final answer will not be a number.

i will present two methods: the former is a more resourceful way but the one i used. the second is the way i used, but it gets more algebraically hairy. (it is the attachment.)
(the more resourceful way)
i found this while checking online to verify my answer. this is not the way i did this question.

we know that  and  . we manipulate our formula for volume so that surface area can appear.



differentiate both sides with respect to surface area, a. use the product rule.



to find dr/da, we can differentiate the area formula with respect to area.



we have found dr/da in terms of radius. now going back to our original work:



note that it wants the rate of change as a function of r. we need to replace a with 4π r² so that everything is in terms of r.



so  . the rate of change of the volume with respect to the area is half the radius.

for every 1 cm² increase of the area, the volume goes up by r/2 m³.

( the way i used)
i cannot include this here because has a character limit for answers. see the attachment if you're interested. i would call this a "brute-force method."

The total would be 40.53