[Solution] Establish a coordinate system such that the vertex of the parabola is on the $y$-axis, and the horizontal axis is set along the span, with the center $M$ as the origin.

Since the vertex of the parabola is at $(0,16)$, its equation is $y=a x^{2}+16$. It also passes through the point $(20,0)$, substituting this into the equation we can find
$$
\begin{array}{r}
a=-\frac{1}{25} . \\
\therefore \quad y=-\frac{1}{25} x^{2}+16 .
\end{array}
$$

When $x= \pm 5$, we get $y=15$.
Therefore, the answer is $(B)$.
$5 \cdot 35$ A parabola has a maximum height of 16 inches and a span of 40 inches. The height of the arc (in inches) at a point $M 5$ inches from the center is
(A) 1 .
(B) 15 .
(C) $15 \frac{1}{2}$.
(D) $15 \frac{1}{3}$.
(E) $15 \frac{3}{4}$.
(20th American High School Mathematics Examination, 1969)