Solution:

The weight of a body at height \( h \) is given by:

\[
W_h = W_0 \left( \frac{R}{R + h} \right)^2.
\]

To find \( h \) when \( W_h = \frac{1}{16} W_0 \):

\[
\frac{1}{16} = \left( \frac{R}{R + h} \right)^2.
\]

Taking the square root:

\[
\frac{1}{4} = \frac{R}{R + h}.
\]

Cross-multiplying gives:

\[
R + h = 4R ; h = 3R.
\]

Thus, the height is \( 3R \).