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?
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?
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.
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.)