The weight of a body at the surface of earth is 18 N. The weight of the body at an altitude of 3200 km above the earth's surface is (given, radius of earth $$R_e = 6400$$ km)

We need the weight at an altitude of 3200 km given that the weight on the surface is 18 N and $$R_e = 6400$$ km.

From the relation between gravitational acceleration at altitude h and at the surface, namely $$g_h = \frac{GM}{(R_e+h)^2}$$ and $$g = \frac{GM}{R_e^2}$$, it follows that the weight at altitude h is given by the formula

$$ W' = W\left(\frac{R_e}{R_e + h}\right)^2 $$

Substituting the values into this expression, we obtain

$$ W' = 18 \times \left(\frac{6400}{6400 + 3200}\right)^2 = 18 \times \left(\frac{6400}{9600}\right)^2 = 18 \times \left(\frac{2}{3}\right)^2 = 18 \times \frac{4}{9} = 8\;\text{N} $$

Hence, the weight at an altitude of 3200 km is 8 N, so the correct answer is Option 4: 8 N.