12.17 A rectangular plot of land with an area of $294 \mathrm{~m}^{2}$ needs to be fenced and then divided into two equal parts by another fence. What should the linear dimensions of the plot be for the total length of the fence to be the smallest?
12.17 Let $x$ and $y$ (Fig. 12.2) (linear dimensions of the plot in meters), then the area of the plot is $x y=294$, from which $y=\frac{294}{x}$. The length of the entire fence

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-384.jpg?height=313&width=484&top_left_y=557&top_left_x=660)

Fig. 12.2 can be expressed by the function

$$
f(x)=3 x+2 y=3 x+\frac{588}{x}=\frac{3 x^{2}+588}{x}
$$

where, by the context of the problem, $x>0$. Further, we have $f^{\prime}(x)=\frac{3 x^{2}-588}{x^{2}}$, from which $\quad f^{\prime}(x)=0 \quad$ when $\quad x=14$ (since $x>0$). If $014$, then $f^{\prime}(x)>0$; therefore, $x=14$ is a point of minimum of the function $f(x)$. As a result, we get $x=14$, $y=21$.

Answer: 14 and 21 m.