Solved: A classical problem in the calculus of variations is to find the shape of a

Chapter 2, Problem 38

(choose chapter or problem)

A classical problem in the calculus of variations is to find the shape of a curve \(\mathscr{C}\) such that a bead, under the influence of gravity, will slide from point A(0, 0) to point \(B\left(x_{1}, y_{1}\right)\) in the least  time. See FIGURE 2.R.7. It can be shown that a nonlinear differential equation for the shape y(x) of the path is \(y\left[1+\left(y^{\prime}\right)^{2}\right]=k\), where k is a constant. First solve for dx in terms of y and dy, and then use the substitution \(y=k \sin ^{2} \theta\) to obtain a parametric form of the solution. The curve  turns out to be a cycloid.

The clepsydra, or water clock, was a device used by the ancient Egyptians, Greeks, Romans, and Chinese to measure the passage of time by observing the change in the height of water that was permitted to flow out of a small hole in the bottom of a container or tank. In Problems 39–42, use the differential equation (see Problems 13–16 in Exercises 2.8)

\(\frac{d h}{d t}=-c \frac{A_{h}}{A_{w}} \sqrt{2 g h}\)

as a model for the height h of water in a tank at time t. Assume in each of these problems that \(h(0)=2\) ft corresponds to water filled to the top of the tank, the hole in the bottom is circular with radius \(\frac{1}{32} \text { in }\),\(g=32 \mathrm{ft} / \mathrm{s}^{2}\), and c = 0.6.