Chapter 4.6, Problem 73E

To determine

(a)

To explain $\upsilon$ approach infinity at the endpoints of the interval $\alpha <\theta <\frac{\pi }{2}$.

To determine

(b)

The plot of ${\upsilon }^2$ as a function of $\theta$ for $\frac{\pi }{6}<\theta <\frac{\pi }{2}$ verify that the minimum occurs at $\theta =\frac{\pi }{3}$.

To determine

(c)

$\upsilon$ is minimized for $\theta$ such that $F(\theta )$ is maximized.

To determine

(d)

Verify that $F\text{'}(\theta )=\cos (\alpha -2\theta )\sec \alpha$ and to show that the maximum value $F({\theta }_O)$ takes place

${\theta }_O=\frac{\alpha }{2}+\frac{\pi }{4}$.

To determine

(e)

Show that ${\upsilon }^2{}_{opt}=\frac{32d^2}{-h+\sqrt{d^2+h^2}}$.

To determine

(f)

Show with graph that for fixed $d$, ${\upsilon }^2{}_{opt}$ is an increasing function of $h$.