Reviewing optimization problems.

1. The Minimum Distance Between
Cars on the Highway or
Optimization Review
Deutsch Autobahn A4
by flickr user davipt

2. ABC is a right-angled triangle with the right-angle at A. P is a
point on the hypotenuse of the triangle. Consider the rectangle
which has one corner at each of A and P and has two of its sides
along AB and AC. Where should P be chosen so as to maximise
the area of the rectangle?

3. A factory makes metal trays as follows. It takes a square sheet of
metal, cuts identical squares from each corner of the sheet and
turns up the four resulting flaps to make the sides of the tray
(perpendicular to the base). How big should these removed
squares be, relative to the size of the original sheet, in order to
make the volume of the tray as large as possible?

4. Car B is 30 miles directly east of Car A and begins moving west
at 90 mph. At the same moment car A begins moving north at 60
mph. What will be the minimum distance between the cars and at
what time t does the minimum distance occur ?

5. Find the point P = (x, 0) on the x-axis which minimizes the sum of
the squares of the distances from P to (0, 0) and from P to (3, 2).

6. Romeo is standing on level ground outside Juliet's house. Juliet
is standing in a window. The bottom of the window is 10 feet
above the ground and the window has height 5 feet. Romeo
wants a good look at Juliet. Where is the best place for him to
stand? Romeo's eye-level is 5 foot above the ground and you can
assume that Juliet is big enough to fill the window. (Getting away
from the literature, what I mean is: where does he stand so that
the window subtends the maximum vertical angle at his eye?)