Helix math problem

Question

Mathematical summary: when unwrapped form the cylinder, one revolution of the helix forms the hypotenuse of a right triangle with circumference of the cylinder as the base. The height at one revolution forms the height of the right triangle. Hence, the arc length of one revolution of the helix can be found with the Pythagorean theorem. The height and the circumference along with the inverse tangent function permits finding the angle of the helix relative to horizontal.

1- The tanks are for petroleum storage. there is a (spill) containment wall that blocks visibility of the bases of the tank. Assume the staircase wraps half way around the tank on the left. If the tank is 60 feet in diameter and 60 feet tall, what is the length of the edge of the staircase touching the tank?

2- what is the angle of the staircase ? does angle seem reasonable?

3- Suppose the angle were 50 degrees. What portion of the circumference would be needed for the staircase to reach the top?

Answer 1

Your statement of the problem appears incomplete.  Here is a general solution.

 

Assume a tank of radius R, height H, and a spiral staircase that goes from the ground to the top of the tank turning through an angle of θ (radians), and rising at an angle of φ (radians) with respect to the ground.

 

Unwrapping the staircase and holding it vertical you will see a right triangle with vertical side H and horizontal side D of:

 

D=θR

 

We now have:

 

tan(φ)=H/D = H/(θR)

 

φ=arctan( H/(θR))

 

You can use the last two equations to solve any problem of this type.