'''
You must manufacture a closed cylindrical can (top and bottom included). You have a fixed amount of material so the total surface area 𝐴 of the can is fixed. 
Find the radius 𝑟 and height ℎ of the cylinder that maximize the enclosed volume 𝑉. What is the relation between ℎ and 𝑟 at the optimum? What is the maximal 
volume 𝑉 sub(max) expressed in terms of the fixed surface area 𝐴?
'''

import numpy as np
from scipy.optimize import minimize

# Fixed surface area
A_fixed = 2000  # You can change this value

# Volume function to maximize (we'll minimize the negative)
def volume(params):
    r, h = params
    return -np.pi * r**2 * h  # Negative for maximization

# Constraint: surface area must equal A_fixed
def surface_area_constraint(params):
    r, h = params
    return 2 * np.pi * r**2 + 2 * np.pi * r * h - A_fixed

# Initial guess
initial_guess = [1.0, 1.0]

# Bounds: radius and height must be positive
bounds = [(0.0001, None), (0.0001, None)]

# Define constraint dictionary
constraints = {'type': 'eq', 'fun': surface_area_constraint}

# Run optimization
result = minimize(volume, initial_guess, bounds=bounds, constraints=constraints)

# Extract optimal values
if result.success:
    r_opt, h_opt = result.x
    V_max = np.pi * r_opt**2 * h_opt
    print(f"Optimal radius: {r_opt:.4f}")
    print(f"Optimal height: {h_opt:.4f}")
    print(f"Maximum volume: {V_max:.4f}")
else:
    print("Optimization failed:", result.message)