Category: GEOMETRY
Geometry is probably the first science to use mathematics. Algebra is the child of geometry and mathematics. Below are listed problems that address aspects of geometry. A principle idea of geometry, used in thermodynamics is: the whole equals the sum of the parts.
Eratosthenes (~200 BC) sought to know the circumference of Earth. Knowing geometry and having observed eclipses of the moon, he believed Earth was a spherical mass. To measure its circumference, he devised a method that used the rays of sunlight that hit Earth which he assumed arrive parallel. Use his data (shown in the figure) to calculate the circumference of Earth.
Prove: (A - B) 2 = A 2 - 2AB + B 2
An easy way to precede is to begin by with a slightly easier task regarding
( A + B )².
The common result was first obtained by Euclid around 300 B.C.
The Pythagorean Theorem, a statement of an algebraic truth regarding the sides and hypotenuse of right triangles, can be proved by use of geometry and the general truth:
"The whole equals the sum of its parts."
The image shows an oil rig which rises 500 feet above sea level and is located 50 miles from shore. We are interested to know if the rig might be seen from a restaurant atop an 800 foot hotel at the shoreline.
Calculate the distance from the rig to shore.
Most of us use this fact without thinking about it. To prove the formula, we use the results of a previous proof.
Pharaoh's Engineers
The Great Pyramid of Egypt was constructed to precise proportions. A hypothesis is that the pyramid was constructed to fit inside an imaginary hemisphere with each of its corners and its peak touching the hemisphere. Suppose the hypothesis were true. Draw a sketch of a pyramid and calculate the angle each face makes with the horizontal plane of the desert.
"Power trains" are the mechanical parts by which the power of combustion is transformed into power of a rotating shaft. A sketch of a simple crank-rod-piston arrangement is shown. Engine designers must know the position of the piston face for every position of the crank. The mathematical tool, vectors, makes this task logical.