related rates of change
I am new to this forum so I hope i post this question at the right level.
A spherical vacuole (something in a cell) is decreasing in size so that its volume decreases at a constant rate of 100 cubic micrometres per seconds. At what rate is the radius decreasing at the instant when the radius of 5 micrometre.
This question needs the chain rule and is based on related rates of change.
So, the [change in the radius wrt time] is equal to the [change in the radius wrt to the volume] * [change in the volume wrt time]
dr /dt = dr/dv * dv/dt
The change in volume wrt time is constant at 100, so can substitute dv/dt for 100 (or -100 because its a decrease?)
dr /dt = dr/dv * 100
Now to find how the radiu changes wrt volume
v = 4/3 * pi * r^3
r = (3/4 * 1/pi * v) ^1/3
when the radius is 5, the volume is 523.6
to find dr/dv, use teh chain rule
r = u^1/3 where u = (3/4 * 1/pi * v)
dr/du = 1/3u^-2/3
du/dv = (3/4 * 1/pi )
= 1/3 (3/4 * 1/pi * v)^-2/3 * (3/4 * 1/pi )
= 1/4 * pi * (3/4 * 1/pi * v)^-2/3
so that's how r changes wrt to the volume. I'm sure that is wrong!
If i then substitute the volume for then the radius is 5
= 1/4 * pi * (3/4 * 1/pi * 523.6 )^-2/3
then multiply * 100 i should have the answer.
But the answer is 1/pi!!!
I have no idea how they get that. All of the previous examples in the book i am using have been based on the approach i've just used.
many thanks for any help
Welcome to MHF!
You're making this way harder than it needs to be.
You are given dV/dt and r, so use that info to solve for dr/dt.