Let the car turn off the highway at a distance x from the...

Let the car turn off the highway at a distance $x$ from the point $D$. So, $C D=x$, and if the speed of the car in the field is $v$, then the time taken by the car to cover the distance $A C=A D-x$ on the highway $$ t_{1}=\frac{A D-x}{\eta v} $$ and the time taken to travel the distance $C B$ in the field $$ t_{2}=\frac{\sqrt{l^{2}+x^{2}}}{v} $$ So, the total time elapsed to move the car from $$ t=t_{1}+t_{2}=\frac{A D-x}{\eta v}+\frac{\sqrt{l^{2}+x^{2}}}{v} $$ For $t$ to be minimum $$ \frac{d t}{d x}=0 \text { or } \frac{1}{v}\left[-\frac{1}{\eta}+\frac{x}{\sqrt{l^{2}+x^{2}}}\right]=0 $$ or $\quad \eta^{2} x^{2}=l^{2}+x^{2} \quad$ or $\quad x=\frac{l}{\sqrt{\eta^{2}-1}}$

Let the car turn off the highway at a distance $x$ from the point $D$. So, $C D=x$, and if the speed of the car in the field is $v$, then the time taken by the car to cover the distance $A C=A D-x$ on the highway
$$
t_{1}=\frac{A D-x}{\eta v}
$$
and the time taken to travel the distance $C B$ in the field
$$
t_{2}=\frac{\sqrt{l^{2}+x^{2}}}{v}
$$
So, the total time elapsed to move the car from
$$
t=t_{1}+t_{2}=\frac{A D-x}{\eta v}+\frac{\sqrt{l^{2}+x^{2}}}{v}
$$
For $t$ to be minimum
$$
\frac{d t}{d x}=0 \text { or } \frac{1}{v}\left[-\frac{1}{\eta}+\frac{x}{\sqrt{l^{2}+x^{2}}}\right]=0
$$
or $\quad \eta^{2} x^{2}=l^{2}+x^{2} \quad$ or $\quad x=\frac{l}{\sqrt{\eta^{2}-1}}$

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Key Concepts

Calculus Optimization

This concept involves finding the minimum or maximum of a function—typically by taking its derivative with respect to a variable and setting the derivative equal to zero. In the problem, the total time is expressed as a function of one variable, and the optimal value that minimizes time is found by solving dt/dx = 0.

Differentiation

Differentiation is used to determine how a function changes as its input changes. Here, the derivative of the total time function with respect to the off-highway distance is computed to identify the point where the time is minimized, indicating an optimal transition point between two different travel speeds.

Path Optimization in Varying Mediums

This concept involves finding the best path when traveling through different regions where movement speeds vary. The approach used here is analogous to finding the minimum travel time path in problems related to Fermat's principle or Snell's law, although applied to a car's journey rather than the path of a light ray.

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