11. Through experimental research, experts have found that the attention index of junior high school students changes with the duration of the teacher's lecture. At the beginning of the lecture, students' interest surges, then there is a period during which their interest remains stable, followed by a period of dispersion. The attention index $y$ changes with time $x$ (unit: $\mathrm{min}$) as shown in Figure 5 (the larger the $y$, the more concentrated the students' attention). When 0 $\leqslant x \leqslant 10$, the graph is part of a parabola. When $10 \leqslant x \leqslant 20$ and $20 \leqslant x \leqslant 40$, the graph is a line segment.
(1) When $0 \leqslant x \leqslant 10$, find the function relationship between the attention index $y$ and time $x$;
(2) A math competition problem requires 24 minutes of explanation. Can the teacher arrange it appropriately so that the students' attention index remains no less than 36 throughout the explanation?
Three, 11. (1) When $0 \leqslant x \leqslant 10$, $y=-\frac{1}{5} x^{2}+\frac{24}{5} x+20$.
(2) When $20 \leqslant x \leqslant 40$, $y=-\frac{7}{5} x+76$.

When $0 \leqslant x \leqslant 10$, let $y=36$, solving gives $x=4, x=20$ (discard);
When $20 \leqslant x \leqslant 40$, let $y=36$, solving gives
$$
x=\frac{200}{7}=28 \frac{4}{7} \text {. }
$$

Since $28 \frac{4}{7}-4=24 \frac{4}{7}>24$, the teacher can, with appropriate arrangements, complete the teaching of this competition problem while the students' attention index is no less than 36.