(Solved): 9. A maple tree is tapped by drilling a hole into the trunk and the sap is collected in a pail with ...

9. A maple tree is tapped by drilling a hole into the trunk and the sap is collected in a pail with base radius 5 inches (as shown in the figure). Let $$h$$ be the height of the sap in the pail and let $$r$$ be the upper radius of the spa collected, both measured in inches. The volume $$V$$ of sap in the pail is changing at a rate of $$\frac{1}{20}$$ cubic inches per minute. The volume $$V$$ of a pail with base radius 5 , height $$h$$, and upper radius $$r$$ is $$V=\frac{?}{3}?h?\left(r^2+5r+25\right)$$ (a) Find an expression for $$\frac{dV}{dt}$$ in terms of $$r,h$$, and their derivatives. (b) Suppose when $$h=3,r=8$$ and $$\frac{dr}{dt}=2?\frac{dh}{dt}$$. Find $$\frac{dh}{dt}$$ and explain the meaning of this answer in the context of the problem. (c) Suppose the pail is replaced by a circular cylinder, so that $$r=5$$ is constant. Find $$\frac{dh}{dt}$$.