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°.

## Set 16

## Illusion of Wrong Areas

## Water Poured Into Liter Cube

Water is poured at a constant rate into a one-liter cube near its corner. The water rises to the halfway mark, stops for a moment, and then continues to rise at exactly half its previous rate. The reason is that inside the cube is a hollow cylindrical can that has been attached to the bottom. What are the height and radius of the can? (Recall that a one-liter cube measures 10 cm on each edge.)

## Marvin and Deedee

A: Marvin tied his dog, Deedee, with a rope 10 meters long at an outside corner of his regular hexagonal walled garden having sides of 8 meters. What is the area of ground that Deedee can cover, as determined by the limits of the rope and the walls?

B: Now do the same problem with Deedee tied inside the garden.

## Midline Triangle Area

In a △ABC, D is the midpoint of AB; E is the midpoint of DB; and F is the midpoint of BC. If the area of △ABC is 96, then the area of △AEF is:

- 16
- 24
- 32
- 36
- 48

## Triangle Trapezoid Area Ratio

Find the ratio of the area of a △RVW to trapezoid STVW.

## Area Sum of Equilateral Triangles 1

Here are two equilateral triangles; one has side *x* and the other *y*. Find the length of the side of an equilateral triangle whose area is the sum of the areas of these two triangles.

## Half Square Foot, Half Foot Square

What part of ½ square foot is ½ foot square? (Your answer will be some fraction.)

## Overlapping Equilateral Triangles

Two congruent equilateral triangles are placed so that their overlap is a regular hexagon. The triangles each have area 24 sq. cm. What is the area of the hexagon?

## Equilateral Octagon

Here is a large plywood square. You are going to cut a triangle off of each corner to make a regular octagon. The square is *a* inches on a side. How many inches should you come in from each corner (*x* in the picture) so that when you cut off the triangles, all eight sides of the resulting octagon will be equal?