Applied Homework #2,  Due Wed Feb 13, 2002

1.

A movie theater has a screen that is positioned 10 feet off the floor and is 25 feet high. The first row of seats is 9 feet from the screen and the rows are 3 ft apart. The floor of the seating area is inclined at an angle of $\alpha=20^o$ above the horizontal and the distance up the incline that you sit is x. The theater has 21 rows of seats, so $0\leq x \leq 60$. Suppose you decide that the best place to sit is where the angle $\theta$ subtended by the screen at your eyes is a maximum. Let us also suppose that your eyes are 4 feet above the floor, as in the figure. (This problem is similar to the problem we solved in Applied homework 1, but this is more complicated and requires the help of technology to solve.)

(a)

Given that

$$\theta = \arccos\left(\frac{a^2 + b^2 - 625}{2ab}\right)$$

where

$$a^2 = (9 + x \cos\alpha)^2 + ( 31 -x \sin\alpha)^2$$

and

$$b^2 = (9 + x \cos\alpha)^2 + ( x \sin\alpha\, - 6)^2$$

use a graph of $\theta$ as a function of x to estimate the value of x that maximizes $\theta$. In which row should you sit? What is the viewing angle $\theta$ in this row?

(b)

Use your CAS (computer algebra system - eg: Mathematica) to differentiate $\theta$ and find a numerical value for the root of the equation $d\theta/dx =0$. Does this value confirm your result in part (b)?

(c)

(optional) Derive the formula given in part (a).

2.

A man initially at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tangent to the curve (see the figure).

(a)

Show that if the path followed by the boat is the graph of the function y=f(x), then

$$f(x)=\frac{dy}{dx}= \frac{-\sqrt{L^2-x^2}}{x}$$

(b)

Determine the function y=f(x).