Project 3. Estimating Arc Lengths.
In aviation it is sometimes useful to estimate a distance between points as the length of a circular arc. This results from navigation methods (search for VOR and DME arc if curious). To estimate on the fly they use what is known as the 60:1:1 approximation. It means that 60 miles from a point a one degree arc is approximately one mile in length. In aviation the distances would be in nautical miles (nm), but the ratio does not change if we use statute miles (the usual type).
Here we will practice using the method to approximate then check why it works.
(a) Using the Ratio.
(i)
(ii)
What is the arclength of 2 degrees at a distance of 30 miles?
(iii)
What is the arclength of 5 degrees at a distance of 30 miles?
(iv)
What is the arclength of 10 degrees at a distance of 20 miles?
(b) Explaining the Ratio.
In four steps we will figure out the length of a one degree arc on a circle with radius 60 miles. Use \(P=2\pi r\) where \(P\) is the perimeter and \(r\) is the radius.
(i)
Calculate the perimeter of a circle with radius 60 miles using the formula
(ii)
Calculate the perimeter of a semi-circle (half circle) with radius 60 miles.
(iii)
Calculate the perimeter of a quarter of a circle with radius 60 miles.
(iv)
Calculate the perimeter of \(1/360\) of a circle with radius 60 miles.
(c)
Next we will consider how accurate this estimate is.
(i)
Compare the result of the arclength of \(1/360\) of a circle from the previous step to the estimation of 1 miles from the 60:1:1. What is the error using 3 decimal places from your calculation above?
(ii)
Calculate the arclength for a 15° arc of a circle with radius 20 miles using \(P=2\pi R\) and using the 60:1:1 approximation. What is the difference between these calculations using 3 decimal places?
(iii)
At 90 mph how long does it take to travel the difference in distance from the previous step?
