# Exercise 1: plan a light for a second headland.
import math

NM = 1852.0
NM_PER_SQRT_M = 3.86 / (NM / 1000)

def geographic_range_nm(light_m, eye_m):
    return NM_PER_SQRT_M * (math.sqrt(light_m) + math.sqrt(eye_m))

def luminous_range_nm(intensity_cd, visibility_nm):
    t = 0.05 ** (1 / visibility_nm)
    lo, hi = 0.01, 200.0
    for _ in range(100):
        mid = (lo + hi) / 2
        if intensity_cd * t ** mid / (mid * NM) ** 2 > 2e-7:
            lo = mid
        else:
            hi = mid
    return (lo + hi) / 2

def intensity_needed_cd(range_nm, visibility_nm):
    lo, hi = 1.0, 1e9
    for _ in range(200):
        mid = math.sqrt(lo * hi)
        if luminous_range_nm(mid, visibility_nm) < range_nm:
            lo = mid
        else:
            hi = mid
    return hi

EYE_M, CLEAR_NM, POOR_NM, GROUND_M = 10, 20, 8, 30      # assumed brief for the second headland

# Part 1: the focal height that gives the clear-weather horizon, and the tower that makes it.
focal_needed = (CLEAR_NM / NM_PER_SQRT_M - math.sqrt(EYE_M)) ** 2
print("focal height needed for", CLEAR_NM, "nm:", round(focal_needed, 1), "m; tower on", GROUND_M, "m of ground:", round(focal_needed - GROUND_M, 1), "m")

# Part 2: the geographic range for towers of 10 to 25 m.
for tower in (10, 15, 20, 25):
    geo = geographic_range_nm(GROUND_M + tower, EYE_M)
    print("tower", tower, "m -> focal", GROUND_M + tower, "m ->", round(geo, 1), "nm", "(meets the brief)" if geo >= CLEAR_NM else "(too short)")

# Part 3: the intensity each weather case needs, and which one decides.
cd_clear = intensity_needed_cd(CLEAR_NM, 10)
cd_poor = intensity_needed_cd(POOR_NM, 2)
print("clear case:", round(cd_clear), "cd | poor case:", round(cd_poor), "cd | decides:", "clear" if cd_clear > cd_poor else "poor")

# Part 4: how much extra intensity buys how much extra range at the standard visibility.
base = max(cd_clear, cd_poor)
for factor in (1, 2, 4, 10):
    print(factor, "x intensity -> nominal range", round(luminous_range_nm(base * factor, 10), 1), "nm")
# focal height needed for 20 nm: 41.4 m; tower on 30 m of ground: 11.4 m
# tower 10 m -> focal 40 m -> 19.8 nm (too short)
# tower 15 m -> focal 45 m -> 20.6 nm (meets the brief)
# tower 20 m -> focal 50 m -> 21.3 nm (meets the brief)
# tower 25 m -> focal 55 m -> 22.0 nm (meets the brief)
# clear case: 109757 cd | poor case: 7024443 cd | decides: poor
# 1 x intensity -> nominal range 31.0 nm
# 2 x intensity -> nominal range 32.9 nm
# 4 x intensity -> nominal range 34.8 nm
# 10 x intensity -> nominal range 37.4 nm