11 - The Case of the Alien Agent—Solution

Summary

TOP SECRET

For the eyes of the Core Cell of Alienwatch only.

Analysis of the freutcaquium layer

As you may know, one of your members anonymously sent us top secret documents giving the calculation by the IPP of the volume of freutcaquium fallout in the Earth's crust. He (or she) pointed out that this calculation did not take into account the curvature of the Earth, and might therefore have overestimated the actual amount of freutcaquium that settled on our planet. If this were true, the amount of freutcaquium might have been insufficient to wipe out the ancient aliens that you (and we) believe may have populated the Earth at that time.

However, our calculations seem to indicate that the error made by the IPP did not cause their results to be incorrect by more than at most 3%, which is probably within the experimental error in the field measurements obtained by the IPP geologists. We present our calculations below, and invite you to verify them.

The layer of freutcaquium can be thought of as a solid of revolution obtained as follows. Consider a circle K of radius R with center at the origin. Above the portion of the circle in the first quadrant consider a curve C with the following property: if 0 ≤ θ ≤ α ≤ π/4, and if L(θ) is a line through the origin O lying in the first quadrant and making an angle θ with the Y-axis, then the distance along L(θ), from O to C is R + s(θ), where s(θ) is defined to be d — (dθ/α) (see Figure A).