942. A wire of length $20 m$ is required to fence a flower bed, which should have the shape of a circular sector. What radius of the circle should be taken so that the area of the flower bed is maximized?
Solution. Let the radius of the circle be $x$, and the length of the arc of the sector be $y$ (Fig. 71). The area of the circular sector is $S=\frac{1}{2} x y$. The function $S$ is to be investigated for a maximum. Note that $S$ depends on two variables, $x$ and $y$. Express $y$ in terms of $x$ (or vice versa). According to the problem, the perimeter of the circular sector is 20 m, i.e., $2 x+y=20$ (the constraint equation), hence $y=2(10-x)$.

Therefore,

$$
\begin{gathered}
S(x)=x(10-x) \\
S^{\prime}(x)=10-2 x, 10-2 x=0, x=5 \\
S^{\prime \prime}(x)=-2, S^{\prime \prime}(5)=-2<0 \text { (max) }
\end{gathered}
$$

Thus, the radius should be $5 \mathrm{~m}$.