Surface area of a sphere?

$\begingroup$

How would I solve the following problem?

Find the rate of change of surface area of a sphere with respect to its diameter D.

I know the formula for surface area of a sphere is

$A=4\pi r^2$

So I know the rate of change of area with respect to radious is $\frac{dA}{dR}$

but how would I find find it with respect to diameter?

$\endgroup$ 2

2 Answers

$\begingroup$

Hint: $r = \dfrac 12 d.\quad$ (The diameter of a sphere is twice the length of the radius.)

That gives you $$A = 4\pi\left(\frac 12 d\right)^2 = 4\pi \left(\frac 14\right) d^2 = \pi d^2$$

Now you can find $\;\dfrac{dA}{dd} = 2\pi d.$

$\endgroup$ 2 $\begingroup$

$A=4\pi r^2=\pi D^2$, so $\frac{dA}{dD}=2\pi D$.

$\endgroup$