Question

From point $A$ located on a highway (shown in figure above) one has to get by car as soon as possible to point $B$ located in the field at a distance $l$ from the highway. It is known that the car moves in the field $\eta$ times slower than on the highway. At a distance $CD=\frac{xl}{\sqrt{{\eta }^2-1}}$ from point $D$ one must turn off the highway. Find $x$.

Answer 1

Let the car turn off the highway at a distance $x$ from the point $D$. So, $CD=x$, and if the speed of the car in the field is $v$, then the time taken by the car to cover the distance $AC=AD-x$ on the highway
$t_1=\frac{AD-x}{\eta v}$ (1)
and the time taken to travel the distance $CB$ in the field
$t_2=\frac{\sqrt{l^2+x^2}}{v}$ (2)
So, the total time elapsed to move the car from point $A$ to $B$
$t=t_1+t_2=\frac{AD-x}{\eta v}+\frac{\sqrt{l^2+x^2}}{v}$
For $t$ to be minimum
$\frac{dt}{dx}=0$ or $\frac{1}{v}[-\frac{1}{\eta }+\frac{x}{\sqrt{l^2+x^2}}]=0$
or ${\eta }^2{x}^2=l^2+x^2$ or $x=\frac{l}{\sqrt{{\eta }^2-1}}$.