Question

Brandon is on one side of a river that is 60 m wide and wants to reach a point 250 m downstream on the opposite side as quickly as possible by swimming diagonally across the river and then running the rest of the way. Find the minimum amount of time if Brandon can swim at 2 m/s and run at 4 m/s.

Answer 1

To minimize the total time, Brandon should swim at an angle that balances the time spent swimming and running.

Step-by-step explanation:

To minimize the total time it takes to reach the point on the opposite side of the river, Brandon should swim at an angle that minimizes the time spent swimming and maximize the time spent running. This can be achieved by considering the vector components of Brandon's velocity.

Let's denote the speed of the river current as v and the width of the river as d. The time spent swimming across the river at an angle can be calculated as d/(2*sin(theta)), where theta is the angle between the river and the resultant velocity vector of Brandon. The time spent running the remaining distance is (250 - d*cos(theta))/4. To find the minimum total time, we need to find the angle theta that minimizes the sum of these two times.

By calculating the derivative of the total time with respect to theta and setting it equal to zero, we can find the angle that minimizes the total time. Once we have the angle, we can plug it back into the time equation to find the minimum total time.