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?
distance of boat rowing = sqrt(5.5^2 + x^2)
distance of running = 6.5 - x
total travel time
t = sqrt(5.5^2 + x^2)/1.2 + (6.5 - x)/3.4
dt/dx = x/(1.2*sqrt(5.5^2 + x^2) - 1/3.4
d^2t/dx^2 = +ve at any x
x is at its minimum when dt/dx=0
x/(1.2*sqrt(5.5^2 + x^2) = 1/3.4
x = 6/sqrt(253) * 5.5 = 2.075 miles
t = 4.899 + 1.302 = 6.201 hours
56
To determine the point downshore where the lighthouse keeper should land to get to the town as quickly as possible, we need to calculate the time it takes for both rowing and running options.
Let's assume the distance from the lighthouse to the landing point is x miles. The remaining distance from the landing point to the town would be 6.5 - x miles.
Using the formula distance = speed * time, we can determine the time it would take for the lighthouse keeper to row and run.
Time taken to row from the lighthouse to the landing point:
Rowing distance = x miles
Rowing speed = 1.2 miles per hour
Rowing time = x / 1.2
Time taken to run from the landing point to the town:
Running distance = 6.5 - x miles
Running speed = 3.4 miles per hour
Running time = (6.5 - x) / 3.4
To minimize the total time, we want to minimize the sum of the rowing time and the running time.
Total time = rowing time + running time
Total time = x / 1.2 + (6.5 - x) / 3.4
We can now minimize the total time by differentiating it with respect to x and setting it equal to zero.
d(total time) / dx = 0
Differentiating the total time equation:
d(x / 1.2 + (6.5 - x) / 3.4) / dx = 0
(1 / 1.2) - (1 / 3.4) = 0
0.833 - 0.294 = 0.539
Therefore, the optimal value for x is approximately 0.539 miles.
The lighthouse keeper should land approximately 0.539 miles downshore from the point on the shoreline nearest the lighthouse to minimize the time it takes to reach the town.