## 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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