10. Foundational Calculus with Python

Activity 10.01: Maximum Circle-to-Cone Volume

Solution:

1. To find the volume of the resulting cone, you need the height of the cone and the radius of the base, as in the figure on the right of Figure 10.33. First, we find the circumference of the base, which is equal to the arc length AB in the cut circle on the left. You can set R to 1 since all we're interested in is the angle.

   Radian measurements make finding arc lengths easy. It's just the angle left over from the cut, which is 2π - θ times the radius R, which we're setting to 1. So θ is also the circumference of the base of the cone. We can set up an equation and solve r:

   Figure 10.34: Formula to calculate the radius

2. We'll code that into our program. We'll need to import a few things from Python's math module and define the r variable:

   from math import pi,sqrt,degrees
   def v(theta):
       r = (2*pi - theta)/(2*pi) 

...