A lighthouse is built on an exposed reef, 5.5 miles off-shore. The shoreline is perfectly straight, and a town is located 6.5 miles downshore from the point on the shoreline nearest the lighthouse.

The lighthouse keeper needs to go from the lighthouse to the town to get fresh supplies. He can row a boat at 1.2 miles per hour, and run at 3.4 miles per hour. How far downshore towards the town should he land, if he wants to get to the town as quickly as possible?

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x=2.075 miles
t=6.200 hours

To find the distance downshore towards the town where the lighthouse keeper should land to reach the town as quickly as possible, we need to consider the combination of rowing and running speeds.

Let's divide the journey into two parts:
1. Rowing from the lighthouse to a point on the shoreline.
2. Running from the shoreline to the town.

Let's denote the distance the lighthouse keeper rows as 'x' and the distance he runs as 'y'.

Therefore, the total time taken for the journey is the sum of the time taken for rowing and running.

1. Time taken for rowing:
Since the lighthouse is 5.5 miles offshore and the rowing speed is 1.2 miles per hour, the time taken for rowing is 5.5/1.2 = 4.583 hours.

2. Time taken for running:
The distance from the shoreline to the town is 6.5 miles. Since the running speed is 3.4 miles per hour, the time taken for running is 6.5/3.4 = 1.912 hours.

Now, the total time for the journey is the sum of the rowing time and running time:
Total time = Time taken for rowing + Time taken for running
Total time = 4.583 + 1.912
Total time ≈ 6.495 hours

To reach the town as quickly as possible, the lighthouse keeper should aim to minimize the total time. Since the total time is a function of x and y, we need to find the appropriate values for x and y.

To minimize the time, we need to differentiate the total time with respect to x and y, and set both derivatives equal to zero. However, since the problem doesn't specify any constraints on the distance x, solving for the optimal values of x and y is not possible with the given information.

Thus, without additional information or constraints, we cannot determine exactly how far downshore towards the town the lighthouse keeper should land to minimize the time taken to reach the town.