Question

From point $A$ located on a highway, one has to get by a car as soon as possible to point $B$ located in the field at a distance $l$ from point $D$. If the car moves $n$ times slower in the field, at what distance $x$ from $D$ one must turn off the highway.

• A. $x=\frac{l}{\sqrt{n^3-1}}$
• B. $x=\frac{l}{\sqrt{n^2-1}}$
• C. $x=\frac{l}{\sqrt{n^3-2}}$
• D. $x=\frac{l}{\sqrt{n^2-2}}$

Answer 1

The correct option is B $x=\frac{l}{\sqrt{n^2-1}}$

$letthedistancebetweenpointsAandDbeytimetakenbythecar=\frac{y-x}{v}+\frac{\sqrt{l^2+x^2}}{v/n}=\frac{y-x}{v}+\frac{n\sqrt{l^2+x^2}}{v}\frac{dt}{dx}=\frac{-1}{v}+\frac{2nx}{2v\sqrt{l^2+x^2}}=0\Rightarrow x=\frac{l}{\sqrt{n^2-1}}$