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?

## Set 17

## 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?

## Medians Triangle Area

In △ABC, medians AD & BE intersect at G and ED is drawn. If the area of △EGD = *k*, find the area of △ABC.

## Overlapping Triangles Make Octagon

Two congruent squares are placed so that their overlap is a regular octagon. The squares each have area 1 sq. meter.

What is the area of the octagon?

## Fence Enclosing Acres

Two miles of fence will enclose a square of 156.25 acres. How large a square pasture will 4 miles of fence enclose?

## 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.)