# Calculate the height of a tower based upon two facts

This is not homework, it's simply something I made up in order to receive answers that will clarify my understanding of how to solve these types of problems myself.

We need the height of the tower. The facts we have:

The circumference of the tower at the bottom level is 300 kilometers. The circumference of the tower reduces gradually as height increases, so that it's like a cone. It can be climbed by walking on a slope that circles around the entire tower from the bottom to the top. If it takes an average man 3500 hours to climb to the top, how high is the tower and at what degree is the slope and so on -- assuming 5 km/h walking speed on LEVEL GROUND.

Any ideas on how one might solve such a problem? Is everything that we need in place? Is more information needed, if so, what information?

• Well, you should find the inverse function for the arc length of the conical spiral: mathworld.wolfram.com/ConicalSpiral.html . Then you can find $h$ knowing $r$ at the base and assuming a constant speed (say 5 kph) and so a length of 16000km. – N74 Aug 27 '16 at 1:15
• You need some relationship between slope and speed, and some information about the ramp - is the angle about the vertical axis proportional to the elevation at that point on the ramp? – mathguy Aug 27 '16 at 1:16
• @N74 that assumes climbing speed is equal to walking speed on level ground (certainly possible but "not in evidence" yet). And how did you get 16000km, I get 17500. – mathguy Aug 27 '16 at 1:17
• It assumed also that, for every complete round of the tower, you will be raised of a constant height. It's up to you to specify what "an average man" is: in a math discussion everything must have a single interpretation. – N74 Aug 27 '16 at 1:25
• Your question is not defined unless you claim the tower is a cone. – Moti Aug 27 '16 at 1:46