Answer:
[tex]\frac{dr}{dh} =-\frac{r}{2h}[/tex]
Step-by-step explanation:
The volume of a cylinder is given by the formula:
V = πr²h,
Where V is the volume, r is the radius of the cylinder and h is the height of the cylinder.
We are to express the rate of change of the radius of the clay with respect to the height of the clay in terms of height h and radius r.
That means we express dr / dh in terms of height h and radius r. To do this, we have to first differentiate the volume with respect to the height, i.e. find dV / dh:
V = πr²h
[tex]\frac{dV}{dh}= \frac{d}{dh} (\pi r^2h)\\\\\frac{dV}{dh}= \pi r^2 + 2\pi rh \frac{dr}{dh} \\\\The\ volume\ is\ constant\ hence\ \frac{dV}{dh}=0.\ We\ get\ that:\\\\0= \pi r^2 + 2\pi rh \frac{dr}{dh}\\\\- \pi r^2 =2\pi rh \frac{dr}{dh}\\\\ \frac{dr}{dh}=-\frac{\pi r^2}{2\pi rh}\\\\ \frac{dr}{dh}=-\frac{r}{2h}[/tex]:
