Given that cos 30° = √3/2 and cos 36° = (1 + √5)/4. Use formulas for the cosines of sums of angles, differences of angles, etc., to come up with an exact expression (involving radicals) for cos 3°.

## Rich Math Problems

## Tape Around a Pipe

A long strip of insulating tape is to be wrapped around a hot water pipe. If the width of the strip is w and the diameter of the pipe is d, at what angle must the strip be placed at the edge of the pipe so that no overlapping and no spaces occur?

## Illusion of Wrong Areas

## Circumference

In a given circle a central angle of 40° intercepts a chord of 6 cm. What is the circumference of the circle?

## Volcano

A new volcano has suddenly appeared where a small strip mall used to be. You are the head of the intrepid surveying crew whose job it is to determine the height of the peak. Traveling bravely toward the volcano, you stop at a safe distance and measure an angle of elevation of 21� to the top. You continue resolutely for another half mile and measure an angle of elevation of 35�. You and your crew had better not get any closer just now — there’s a new spew of ash coming out from the peak. So retreat, do some calculating, and tell the waiting reporters just how tall the volcano is. (They’ll want the answer in feet, of course.)

## The Lighthouse

You are sailing your splendid yacht, the Gemini, down the Hudson River into New York Harbor. It is a sunny afternoon, with a light breeze, and everything is going well. Your crew has been scrubbing down the bulkheads, polishing the brass hardware, and oiling up the teakwood decks. Your chef has begun preparing a splendid dinner on the afterdeck for you and your guests — life is good.

You are approaching the great gray George Washington Bridge, which spans the Hudson from Manhattan Island to New Jersey. Beneath the bridge, on the Manhattan side, is a famous little red lighthouse. You take a sighting of the lighthouse and observe that it is 15 degrees to port. You proceed for two minutes at your stately rate of 5 knots, and you observe that the lighthouse is now 29 degrees to port. How close will you come to the lighthouse as you pass under the bridge?

(Data: 1 knot is approximately 6076 feet per hour.)

## Rocket Launch

You are 2 km from the launch pad, watching a mighty rocket climb skyward, under the fearless command of the intrepid Captain Knight. Your first observation gives an angle of elevation of 21 degrees. Five seconds later, the angle increases to 35 degrees. What was the rocket’s average speed during the five-second interval?

## Rhombi, Locus of Vertex

and are noncollinear rays (endpoint A). Rhombi are drawn, having one vertex at A, one vertex on , and one vertex on . What is the locus of the fourth vertex?

## Triangle Rotating in Square

Equilateral triangle EKP with side EK of length 2 inches is placed inside square EAMI with side of length 4 inches so that K is on side EA. The triangle is rotated clockwise about K, then P, and so on along the sides of the square until E, K, and P return to their original positions. The length of the path in inches traversed by vertex P is equal to:

- 20/3
- 32/3
- 12
- 40/3
- 15

## Crumbs on a Table

A maid was about to wipe the crumbs off a rectangular table measuring 4 ft. by 8 ft. when the butler, a methodical type, took over. “You need a system!” he said. They both had very short arms and could only reach 2 feet with the dust cloth, so the butler decided to get all the crumbs together at center point C along one side. He walked along the opposite side, AB, pushing all the crumbs as far as he could directly toward C, and then did the same thing walking down each end; finally he went to the other long side to pull everything toward the dustpan. When he finished, the maid complained. Why?