## 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?

## Volume of a Shell and Solid

Imagine a rectangular solid that is made up of smaller unit cubes. The solid measures x by y by z. Now imagine building a shell of unit cubes around the entire solid.

First question: what is an equation for the number of unit cubes in the shell, in terms of x, y, and z?

Second question: for what x, y, and z is the number of cubes in the rectangular solid equal to the number of cubes in the shell?

## Geometric Mean in Parallelogram

Given parallelogram ABCD with its diagonals drawn. A line through B intersects AC at E, DC at G, and line AD at F. Observe that △AEF is similar to △CEB, and show that EB is the geometric mean of EG and EF.

## Humber Bridge, Radius of Earth The single-span suspension bridge crossing the estuary of the Humber River on the east coast of England has towers that are 162.5 meters tall (AC=BD). The span at sea level is 1410 meters (about 4/5 of a mile). The tops of the towers are 3.6 cm farther apart than the bases (i.e. AB=1410.036 m). What is the radius of the earth?

## City Hall Mouse, Numerical

City Hall has a splendid rectangular black-and-white tiled floor. The tiles are squares. There are 93 of them in one direction and 231 in the other. A mouse runs in a straight line diagonally from one corner of the floor to the opposite corner. How many tiles does it cross?

## Seven Congruent Rectangles You see a rectangle that has been made by putting seven congruent rectangles together. The perimeter of the large rectangle is 114 cm. What is its area?

## Height of Octagonal Stop Sign Each of the eight sides of this stop sign is 12.5 inches long. How tall is the sign, from top to bottom? (It’s a regular octagon.)