Answer
If the distance of the lamp is raised from 15 inches to 30 inches, the amount of illumination fades to the factor of $\frac{1}{900}$.
Work Step by Step
If the distance d of the lamp over the desk is 15 inches, then the square of the distance is
$\begin{align}
& {{d}^{2}}={{\left( 15 \right)}^{2}} \\
& {{d}^{2}}=225 \\
\end{align}$
The illumination l from a light source varies inversely as the square of the distance d.
Thus,
$\begin{align}
& l=\frac{k}{{{d}^{2}}} \\
& =\frac{k}{225}
\end{align}$.
So,
$l=\frac{k}{225}$ , where, k is the constant of variation.
Now, if the distance of the lamp is raised to 30 inches, it changes inversely the amount of illumination as:
$\begin{align}
& l=\frac{k}{{{d}^{2}}} \\
& =\frac{k}{{{30}^{2}}} \\
& =\frac{k}{900}
\end{align}$
Thus, the amount of illumination fades to the factor of $\frac{1}{900}$.
