What are the applications of modulus of elasticity?

As the term implies, “Modulus of Elasticity relates to the elasticity or “flexibility” of a material. The value of modulus of elasticity is very much significant relating to deflection of certain materials used in the construction industry. Take for example the general E value of mild carbon steel is about 200 GPA compared to about 70 GPA for aluminum. This simply translate that aluminum is 3 times flexible than steel.

Why nautical mile is different from statute mile?

One nautical mile is defined by one latitude minute of arc (there are 60 such minutes to a degree). This equals 1852 meters, and roughly (but coincidentally) 2000 yards or 6000 feet. (Edit: actually, a standard nautical mile is 6076 feet, 6000 feet and 2000 yards are commonly used approximations, but produce an error of about 1%).
The statute mile had a little fuzzier definition to start with, as one mile was the same as 1000 roman PAces/steps. The definition has since changed, but one statute mile equals about 1609 meters.

What is the tensile strength of wood?

The tensile strength of a material is the value at which the material fails when subjected to a tensile force. (Tensile means a force pulling the wood fibers lengthwise, as opposed to a compressive force.) There are a lot of different types and load cases for wood, so there is not one answer to this question. For example,
wood used outside will fail at a lower load than wood inside. Wood is also anisotropic, i.e. it has different strength in different directions.
Example values of tensile strength vary anywhere from 175 pounds per square inch for Utility Grade pine to 1400 PSI for Dense Select Structural grade Douglas Fir. A
good reference for material strength data for most species and grades of commercially available wood can be found in the American Institute of Timber Construction handbook.

What distance in meters on the ground is the equivalent of one second of arc in longitude or latitude?

One minute of arc as measured at the centre of the Earth covers one nautical mile on the surface of the Earth at mean sea level. One nautical mile is 6080 feet or 1853.2 meters. Therefore one second of arc would be 6080 / 60 = 101.3 feet or 30.886 meters.
Lines of latitude are at regular intervals PArallel to the equator. The relationship between degrees of latitude and the distance sPAnned on the earths surface remains constant. Therefore at all latitudes 1 minute of latitude sPAns 1 nautical mile on the earths surface.
Lines of longitude converge at the poles. Therefore, the relationship between degrees of longitude and the distance sPAnned on the earth�s surface is reduced as the poles are approached.
At the equator, the distance sPAnned by 1 minute of longitude would be 1 nautical mile. At the poles, it would be zero. To calculate the actual distance on the surface of the earth between two points of known latitude and longitude requires knowledge of spherical trigonometry to calculate the great circle distance between the two points.
The distances quoted are for the surface of the earth at mean sea level. Distances will be increased above sea level and reduced below it.

At what point does a circle start losing area when being flattened?

The area ratio decreases slowly at first then more rapidly.
The area of an ellipse is A=pi x a X b, a and b are the 1/2 major and 1/2 minor axis. If a = r = b the area is A=pi X r2, which is a circle. Not hard to graph this but you will have to write the area of the ellipse in terms of the circumference.

Did any of the Egyptian pyramids ever fall down?

Yes, many did. The great pyramid and others that have survived did so for a reason. Their shapes made them sturdier over time and were not the first design. Many steppe pyramids were built but were too steep and as the corners eroded, the whole pyramid fell under its own weight. They were built too steep and did not have the base structure as seen in the great pyramid.