15. As shown in the figure, two factories $A, B$ are $0.6 \mathrm{~km}$ apart, and the substation $C$ is $0.5 \mathrm{~km}$ away from both $A$ and $B$. It is planned to lay power lines, first from $C$ along the perpendicular to $A B$ to $D$, then connect to $A, B$. Where should point $D$ be located to minimize the length of the power lines?
15. Let $C D \perp A B$, with the foot of the perpendicular being $E, D E$ having a length of $x \text{ km}$, and given $A B=0.6, A C=0.5$, we get $A E=E B=0.3, C E=$ $\sqrt{(0.5)^{2}-(0.3)^{2}}=0.4, C D=0.4-x, A D=B D=\sqrt{x^{2}+(0.3)^{2}}$. The total length of the power line $l=$ $2 \sqrt{x^{2}+(0.3)^{2}}+0.4-x$, and let $l'=\left[2 \sqrt{x^{2}+(0.3)^{2}}+0.4-x\right]'=\frac{2}{2} \cdot \frac{2 x}{\sqrt{x^{2}+0.09}}-1=$ $\frac{2 x-\sqrt{x^{2}+0.09}}{\sqrt{x^{2}+0.09}}=0$, which means $2 x-\sqrt{x^{2}+0.09}=0$, solving for the unique critical point $x=\frac{\sqrt{3}}{10} \approx 0.17(\text{ km})$.
This practical problem has a minimum value for $t$, so point $D$ should be selected on the perpendicular bisector of $A B$, $0.17 \text{ km}$ away from the midpoint of $A B$.