Question

The radius of a sphere decreases at a rate of 6 m/sec. find the rate at which the surface area decreases when the radius is 20 m. recall that the surface area of a sphere of radius r is given by s = 4πr2

Answer 1

The radius of the sphere decreases at the rate of 6 m/ sec

That is, $\frac{dr}{dt}=-6$

Surface Area of the sphere of radius r is given by $S=4\pi r^2$

Differentiating with respect to t,

$\frac{dS}{dt}=4\pi (2r)\frac{dr}{dt}=8\pi r\frac{dr}{dt}$

$\frac{dS}{dt}=8\pi (20)^2(-6)=-960\pi \: m^2/sec$