To solve this problem, we need to find the optimal path that minimizes the time it takes for Brandon to cross the river and then run along the shore to reach his destination. Here’s a breakdown of the solution:

1. Define the path:

   • Brandon is 50 meters from the opposite shore and needs to reach a point 300 meters downstream.
   • He will swim diagonally across the river to a point $x$ meters downstream from his starting position on the opposite shore, and then run the remaining distance $(300 - x)$ meters.

2. Set up time expressions:

   • Swimming time: The distance Brandon swims is the hypotenuse of a right triangle with legs 50 m (the river width) and $x$ m (the downstream distance he swims).

     • Swimming distance $d_{\text{swim}} = \sqrt{50^2 + x^2}$
     • Time to swim this distance: $t_{\text{swim}} = \frac{\sqrt{50^2 + x^2}}{2}$

   • Running time: After swimming, Brandon needs to run $(300 - x)$ meters.

     • Time to run this distance: $t_{\text{run}} = \frac{300 - x}{4}$

3. Total time:

   • Total time $T(x)$ as a function of $x$: $T(x) = \frac{\sqrt{50^2 + x^2}}{2} + \frac{300 - x}{4}$

4. Minimize the time:

   • To find the minimum time, take the derivative of $T(x)$ with respect to $x$, set it to zero, and solve for $x$.
   • This will give us the optimal point where Brandon should exit the water to minimize his total time.

Let's go through the calculations:

Step 1: Differentiate $T(x)$

The derivative of $T(x)$ with respect to $x$ is: $\frac{dT}{dx} = \frac{x}{2\sqrt{50^2 + x^2}} - \frac{1}{4}$

Step 2: Set the derivative to zero

$\frac{x}{2\sqrt{50^2 + x^2}} = \frac{1}{4}$ Solving this equation for $x$ will give us the optimal distance downstream for Brandon to reach the opposite shore.

Step 3: Calculate and substitute

Let me calculate this for the minimum time.The optimal distance downstream for Brandon to reach the opposite shore is approximately:

$x = \frac{50 \sqrt{3}}{3} \approx 28.87 \text{ meters}$

Step 4: Calculate the minimum time

With $x \approx 28.87$ meters, we can substitute this value back into the time equation to find the minimum time it would take for Brandon to reach his destination. Let’s proceed with that calculation.The minimum time it would take for Brandon to reach his destination is approximately:

$T_{\text{min}} \approx 96.65 \text{ seconds}$

This is the shortest time for Brandon to cross the river and reach the point 300 meters downstream.

Would you like further details on this calculation?

Here are some additional related questions:

1. What if Brandon could swim faster, say at 3 m/s? How would that affect his path?
2. How would the optimal point change if the river width were increased to 100 meters?
3. What if Brandon wanted to minimize his swimming distance rather than time? What path should he choose?
4. How does the swimming and running speed ratio influence the crossing point?
5. Can we derive a general formula for the optimal crossing point for any river width and speeds?

Tip: When optimizing paths with two modes of travel, finding a balance between time and distance is key. Always check the derivative of total time for a minimum time solution.