How does this radius come out 6.2

I'm trying to find out a critical point by solving for r
-4000/r^2 + (16pi*r)/3 = 0
I solved for r by getting the cube root of 12000/16pi and got 13.3
The correct answer was 6.2

You should be able to rearrange it to
r^3 = 12000/16/pi
and if you don't get 6.2 you're doing something wrong on your calculator.

That's because you calculated ∛(12000/16π) instead of (∛12000/(16π))

This is why a good understanding of order of operation is so important

∛(12000/16π) = ∛(750π) = ∛(2356.194490192) = 13.306700395

∛(12000/(16π)) = ∛(12000/50.265482457) = ∛(238.732414638) = 6.203504909

Mαthmφm

i think you just made a calculation error, i inputted the cute root of 1200/16pi and i got 6.2> :/ i think you put cube root of 12000pi/16 on your calculator>