Numerical: maximum cell radius using Hata model

The maximum cell radius is the distance at which Hata path loss equals the largest loss the link can tolerate. Antenna gains and cable losses are not given, so they are taken as 0 dB.

Step 1: Maximum allowed path loss

Pt = 50 W = 10 log(50 000 mW) = 46.99 dBm
Pr,min = −91 dBm

Lmax = Pt − Pr,min = 46.99 − (−91) = 137.99 dB

Step 2: Mobile antenna correction (medium city)

fc = 900 MHz, log 900 = 2.9542, hre = 1 m:

a(hre) = (1.1 log fc − 0.7)hre − (1.56 log fc − 0.8)
       = (3.2497 − 0.7)×1 − (4.6086 − 0.8)
       = 2.5497 − 3.8086
       = −1.26 dB

Step 3: Hata equation (urban)

hte = 30 m, log 30 = 1.4771:

L50 = 69.55 + 26.16 log fc − 13.82 log hte − a(hre)
      + (44.9 − 6.55 log hte) log d
    = 69.55 + 77.28 − 20.41 + 1.26
      + (44.9 − 9.68) log d
    = 127.68 + 35.22 log d

Step 4: Solve for d

127.68 + 35.22 log d = 137.99
log d = 10.31 / 35.22 = 0.2927
d = 10^0.2927 = 1.96 km

Step 5: Cell area

For a hexagonal cell of radius R:

A = (3√3/2)·R² = 2.598 × 1.962² = 10.0 km²

(If a circular cell is assumed, A = πR² = 12.1 km².)

Answer: Maximum cell radius ≈ 1.96 km; hexagonal cell area ≈ 10.0 km² (≈ 12.1 km² if circular). All parameters are within Hata's range (hte 30–200 m, hre 1–10 m, d 1–20 km).