To find the distance RM that minimizes the time taken for a car at location Q on the highway to reach a point P in the field, we need to consider the speeds and distances involved.

Step-by-Step Solution

1. Let's denote:
   • The speed of the car on the highway as v.
   • The speed of the car in the field as \frac{v}{2}.
   • The distance QM = x.
   • The distance RM = y.
2. According to the given figure:
   • The total distance to be traveled in the field is \sqrt{y^2 + d^2}, using the Pythagorean theorem because P forms a right triangle with R and M.
3. The total time taken to reach point P is given by:
   T = \frac{x}{v} + \frac{\sqrt{y^2 + d^2}}{v/2}
   Simplifying, we have:
   T = \frac{x}{v} + \frac{2\sqrt{y^2 + d^2}}{v}
4. To minimize T, differentiate with respect to x and set the derivative to zero:
   \frac{dT}{dx} = \frac{1}{v} - \frac{2x}{v\sqrt{x^2 + d^2}} = 0
   Solving, we find:
   1 = \frac{2x}{\sqrt{x^2 + d^2}}
   Square both sides:
   x^2 + d^2 = 4x^2
   3x^2 = d^2
   x = \frac{d}{\sqrt{3}}
5. Thus, the distance RM that minimizes the time to reach P is:
   RM = \frac{d}{\sqrt{3}}
   Therefore, the correct answer is
   $\frac{d}{\sqrt{3}}$
   .

Conclusion

Therefore, the optimal distance RM for minimizing time to reach the point P is \frac{d}{\sqrt{3}}.