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?

Continue reading

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?

Continue reading