4. On a vast meadow, a lamb is grazing, tied with two ropes to two stakes (each stake with its own rope).

a) What shape will the area on the meadow be, where the lamb can eat the grass?

b) In the middle between the stakes, a rose grows, and the distance between the stakes is 20 meters. What should the lengths of the ropes be so that the lamb cannot eat the rose?
Answer: b) Exactly one of the ropes is shorter than 10 m.

Solution. a) The desired figure is the intersection of two circles with centers at the stakes and radii equal to the lengths $l_{1}$ and $l_{2}$ of the first and second ropes, respectively. Therefore, it can be either the figure shaded in Fig. 3, or a complete circle. The latter case occurs if $\left|l_{1}-l_{2}\right| \geqslant r$, where $r$ is the distance between the stakes, since then one of the circles is completely inside the other, and the desired figure coincides with the circle of the smaller radius.

b) For the rose not to fall within the area on the meadow where the lamb can eat grass, it must lie outside one of the circles bounding this figure. The distance from it to each of the stakes is $r / 2=20 / 2=10 \mathrm{m}$, so the length of one of the ropes must be less than $10 \mathrm{m}$.