Example 8. Given that $a, b, c, d, e$ are real numbers satisfying $a+b+c+d+e$ $=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$. Determine the maximum value of $e$. (7th American High School Mathematics Examination)
Let the plane $\pi: a+b+c+(d+e-8)=0$, and the sphere $O: a^{2} +b^{2}+c^{2}=16-d^{2}-e^{2}$. From Theorem 2 (2), (3), we have
$$
\frac{|d+e-8|}{\sqrt{3}} \leqslant \sqrt{16-d^{2}-e^{2}},
$$

which simplifies to $2 d^{2}+(e-8) d+2 e^{2}-8 e+8 \leqslant 0$, a quadratic inequality in $d$. Therefore,
$$
\begin{array}{l}
\quad(e-8)^{2}-16\left(e^{2}-4 e+4\right) \geqslant 0,15 e^{2}-48 e \leqslant 0 . \\
0 \leqslant e \leqslant \frac{16}{5}, e_{\max }=\frac{16}{5} .
\end{array}
$$