You can run at a speed of 4 mph and swim at a speed of 2 mph and are located on the shore, 3 miles east of an island that is 1 mile north of the shoreline. How far (in mi) should you run west to minimize the time needed to reach the island?

If you run west for x miles, then you have to swim √((3-x)^2+1) miles

So, you want to minimize the total time, which is thus

x/4 + √((3-x)^2+1)/2

To minimize the time needed to reach the island, you should take the shortest path, which can be achieved by forming a right triangle with the shore, the island, and the path you will take.

The distance you need to cover in the water is the hypotenuse of the right triangle, which can be found using the Pythagorean theorem:

Distance in water = √((3 Mi)² + (1 Mi)²)
= √(9 + 1)
= √10 Mi

Since you can swim at a speed of 2 mph, the time needed to cover this distance is:

Time in water = Distance in water / Swim speed
= √10 Mi / 2 mph
= (√10) / 2 hours

To find the distance you should run west, you need to consider the remaining distance needed to reach the island from the shoreline. This distance can be found by subtracting the horizontal distance covered while swimming from the total distance between the shore and the island:

Distance remaining on land = (3 Mi) - (√10 Mi)
= 3 Mi - (√10) Mi

Since you can run at a speed of 4 mph, the time needed to cover this distance is:

Time on land = Distance remaining on land / Run speed
= (3 - √10) Mi / 4 mph
= (3 - √10) / 4 hours

To summarize, you need to run a distance of (3 - √10) miles west to minimize the time needed to reach the island.

To minimize the time needed to reach the island, you need to find the shortest total distance that combines running west and swimming north. Let's break down the problem step by step:

1. Draw a diagram: Draw the shoreline as a straight line, and mark the starting position on the shore. Then, draw a point 3 miles east and 1 mile north as the island. Connect these two points with a straight line to represent the straight-line distance between them.

2. Determine the distance traveled while running: Since you can run at a speed of 4 mph, the time it takes you to run west depends on the distance you cover. Since you want to minimize the total time, you should minimize the distance you run. To do so, you need to determine the shortest distance between your starting position and the point directly south of the island.

3. Calculate the distance to run west: Since you want the shortest straight-line distance between two points, you can use the Pythagorean theorem to find this distance. In this case, the shortest distance can be found by applying the theorem to the right triangle formed by your starting position, the island, and the point where you will run to on the shore. The legs of the triangle are 1 mile (north) and 3 miles (east), so the distance to run west can be calculated as the square root of ((3 miles)^2 - (1 mile)^2).

4. Evaluate the answer: Calculate the distance to run west by taking the square root of (3^2 - 1^2). This equals the square root of (9 - 1), i.e., the square root of 8 miles or approximately 2.83 miles.

Therefore, to minimize the time needed to reach the island, you should run approximately 2.83 miles west along the shore.