Differentials of spherical surface area and volume

[iScience]

please tell me if i did this correctly:

task: I'm trying to divide the differential dA by dV

where.. dA = differential surface area of a sphere, dV = differential volume of a sphere

dA=8$\pi$rdr
dV=4$\pi$r²dr

so then dA/dV= 2/r


Also, if i treat this as a derivative, then would dA/dV = 1/dr?

 

[Simon Bridge]

dA/dV would be how the area changes with the volume.
So you can check it out from the known relations:

$A(R)=4\pi R^2$ is the surface area of a sphere radius R, and
$V(R)=\frac{4}{3}\pi R^3$ is it's volume, then

So find A(V) and differentiate.

 

[lurflurf]

dA/dV= 2/r
this is right
dA/dV = 1/dr
this is not, how did you arrive at it?

 

[HallsofIvy]

For a sphere of radius r, $A= 4\pi r^2$ and [itex]V= (4/3)\pi r^3[itex].

So $dA/dr= 8\pi r$ and $dV/dr= 4\pi r^2$.

Then $dV/dA= (dV/dr)/(dA/dr)= (4\pi r^2)/(8\pi r)= (1/2)r$
and $dA/dV= (dA/dr)/(dA/dr)= (8\pi r)/(4\pi r^2)= 2/r$

"dA/dV= 1/dr" makes no sense because you have a differential on the right side and not on the left.