Chapter 4.5, Problem 73E

To determine

A basketball player stands $d$ feet from the basket let $h$ and $\alpha$ ball $\theta$ ${\upsilon }^2=\frac{16d}{co{s}^{2}q(tanq-tana)}$

a)

$\alpha <\theta <\frac{\pi }{2}$ why does $\upsilon$ approach infinity at the endpoints of this interval

To determine

b)

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

To determine

c)

Set $F(\theta )={\cos }^2\theta (\tan \theta -\tan \alpha )$ why $\upsilon$ is minimized for $\theta$ such that $F(\theta )$ is maximized

To determine

d)

Verify that $F\text{'}(\theta )=\cos (\alpha -2\theta )\sec \alpha$ additional formula used maximum value $F$

To determine

e)

For a give $\alpha$ the optimal angle for shooting the basket ${\theta }_o$ cause it minimizes ${\upsilon }^2$

To determine

f)

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