May 2000
Part 1:
Let y denote latitude in radians, so that
y=0 at the equator and y=pi/2 at the pole.
Let x denote longitude in radians, with x=0 at prime meridian.
Let t be total distance travelled, measured in radii.
Then dy/dt = cos(pi/4) = 1/sqrt(2).
So t=y*sqrt(2)+C, and initial conditions give C=0.
When we finish, y=pi/2, so t=pi/sqrt(2) radii.
Use (25000 miles = 2 pi radii) to conclude
t = 6250 sqrt(2) miles = 8839 miles.
Part 2:
dx/dt = (sin(pi/4))*(1/cos(y)).
Then dx/dy=1/cos(y).
x = log(|sec y + tan y|) + C.
Again initial conditions imply C=0.
We can solve to find y=(pi/2)-2*arctan(exp(-x)).
Substituting x=2*pi we find
y=(pi/2)-2*arctan(exp(-2*pi)),
so that the distance from the pole is:
(pi/2)-y=2*arctan(exp(-2*pi)) = 0.003735 radians = 14.86 miles.
Part 3:
If we change the heading to north by northwest, we find:
dy/dt = cos(pi/8).
dx/dt = (sin(pi/8))*(1/cos(y)).
dx/dy = (tan(pi/8))*(1/cos(y)).
x = (tan(pi/8))*(log(|sec y + tan y|)) + C.
y = (pi/2)-2*arctan(exp(-x/tan(pi/8))).
Recalling tan(pi/8)=sqrt(2)-1, and substituting x=2*pi, we find:
y = (pi/2)-2*arctan(exp(-2*pi/(sqrt(2)-1))),
so that the distance from the pole is:
(pi/2)-y=2*arctan(exp(-2*pi/(sqrt(2)-1)))=0.0000005167 radians
= 0.002056 miles = 10.855 feet.
Part 4:
I was going to make some lame joke about Alfred Hitchcock's movies, "North by Northwest" and "The Birds", but the punch line would have involved penguins at the north pole (erroneously).
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