missile height with a sextant

Here is a clever little problem from my college days. Of course, I have altered it a little for this board.

Bebe Liang has been sent to North Korea by the C.I.A. to determine the size of their new ballistic missile. All she has is a calculator and a sextant, which is a device used to measure the altitude of objects in degrees of arc. The horizon is zero while straight up is 90 degrees. Sailors used them for navigation before GPS.

Bebe reaches the North Korean missile base without being seen. There is a high fence all around the base with rolls of barbed wire at the top. Bebe spots the new missile which is standing vertically, and takes a reading with her sextant at the fence from ground level. The top of the missile is 35.2 degrees above the ground.

Now at this point, Bebe doesn’t know how far away the missile is, so she can’t determine how tall it is. What to do? She paces off 40 feet from the fence. She still doesn’t know how far she is from the missile, she just knows she is 40 feet FARTHER then she was before. Bebe takes another reading with her sextant from ground level, and the top of the missile is now only 26.4 degrees above the ground. So ……..

How tall is the missile?

How far is it inside the fence?

Next Bebe sent back the information using a secret code. She assigned a (different) letter of the alphabet to each of the digits 0,1,2,…,9. Then she sent the message in the form of an addition problem.

..SEND
+MORE
MONEY

She was confident that the CIA would be able to find the unique solution, and discover the answer in the first line.

Of course, if the CIA sends more money, Bebe can buy herself an expensive laser rangefinder.