This optimization problem requires determining the shortest time for the rancher to travel from his ranch to the city using two distinct travel speeds: 40 mph across the open range and 50 mph along the highway.

Let's break this down step by step:

Variables and Equations:

1. Diagram Information:

   • Distance across the range: Hypotenuse of a triangle with legs of 20 miles (north-south) and $x$ miles (east-west).
   • Distance along the highway: $60 - x$, where $x$ is the portion of the eastward highway covered by traveling across the range.

2. Travel Speeds:

   • Across the range: 40 mph
   • Along the highway: 50 mph

3. Objective: Minimize total travel time $T$, which is the sum of the time taken across the range and along the highway.

4. Distance Across the Range: The hypotenuse $c$ is given by: $c = \sqrt{20^2 + x^2} = \sqrt{400 + x^2}$

5. Time Across the Range: $\text{Time} = \frac{\text{Distance}}{\text{Rate}} = \frac{\sqrt{400 + x^2}}{40}$

6. Time Along the Highway: $\text{Time} = \frac{\text{Distance}}{\text{Rate}} = \frac{60 - x}{50}$

7. Total Time: $T(x) = \frac{\sqrt{400 + x^2}}{40} + \frac{60 - x}{50}$

Optimization:

1. Minimize $T(x)$: Take the derivative of $T(x)$ and solve for $x$ where $T'(x) = 0$.

   $T'(x) = \frac{1}{40} \cdot \frac{x}{\sqrt{400 + x^2}} - \frac{1}{50}$

   Simplify: $T'(x) = \frac{x}{40\sqrt{400 + x^2}} - \frac{1}{50}$

2. Set $T'(x) = 0$: $\frac{x}{40\sqrt{400 + x^2}} = \frac{1}{50}$

   Multiply through by $40\sqrt{400 + x^2} \cdot 50$: $50x = 40\sqrt{400 + x^2}$

   Square both sides: $2500x^2 = 1600(400 + x^2)$

   Expand: $2500x^2 = 640,000 + 1600x^2$

   Solve for $x^2$: $900x^2 = 640,000$

   $x^2 = \frac{640,000}{900} = 711.11$

   $x = \sqrt{711.11} \approx 26.68$

3. Interpretation:

   • The rancher should travel approximately 26.68 miles east across the open range before switching to the highway.

Total Time:

Substitute $x = 26.68$ into $T(x)$ to find the minimum travel time: $T(x) = \frac{\sqrt{400 + (26.68)^2}}{40} + \frac{60 - 26.68}{50}$

This completes the solution process. Let me know if you’d like further explanation or calculations!

Related Questions:

1. How does the travel time vary if the highway speed increases to 60 mph?
2. What happens if the rancher takes a direct diagonal path to the city?
3. How would the calculation change if the open-range speed decreases to 30 mph?
4. What optimization techniques could be used to confirm the solution?
5. Could this problem be solved graphically instead of algebraically?

Tip:

Always visualize the geometry of optimization problems to better understand the variables and constraints involved.