(minimum commuting time) A lighthouse lies 2 miles offshore directly across from point A of a straight coastline. The lighthouse keeper lives 5 miles down the coast from point A.What is the minimum time it will take the lighthouse keeper to commute to work, rowing his boat at 3 miles per hour, and walking at 5 miles per hour?

Let the distance from A to his landing place be x

then from there it would be another 5-x miles to his house.
let the distance he rows be d
d^2 = x^2 + 2^2
d = (x^2 + 4)^(1/2)

time for the rowing part = (x^2 + 4)^(1/2) /3
time for the walking part = (5-x)/5

Time = (x^2 + 4)^(1/2) /3 + (5-x)/5
d(Time)/dx = (1/6)(x^2 + 4)^(-1/2) (2x) - 1/5
x/(3√(x^2+4) ) - 1/5
= 0 for a minimum of Time.

x/(3√(x^2+4) ) - 1/5
5x = 3√(x^2 + 4)
25x^2 = 9(x^2+4)
25x^2 = 9x^2 + 36
16x^2 = 36
4x = 6
x = 6/4

sub that into Time = .. and you got it
( I got 1 hour and 32 minutes)
check my arithmetic

To find the minimum commuting time for the lighthouse keeper, we need to consider the time it takes for him to row to the lighthouse and the time it takes for him to walk from his house to point A.

First, let's find the time it takes for the lighthouse keeper to row to the lighthouse. The distance from the coastline to the lighthouse is 2 miles, and he rows at a speed of 3 miles per hour. We can use the formula time = distance / speed to find the time it takes him to row:

Time to row = 2 miles / 3 miles per hour = 0.67 hours (rounded to two decimal places)

Next, let's find the time it takes for the lighthouse keeper to walk from his house to point A. The distance from his house to point A is 5 miles, and he walks at a speed of 5 miles per hour. Again, we can use the formula time = distance / speed to find the time it takes him to walk:

Time to walk = 5 miles / 5 miles per hour = 1 hour

Now, we add the time it takes to row to the time it takes to walk to find the total commuting time:

Total commuting time = Time to row + Time to walk = 0.67 hours + 1 hour = 1.67 hours

Therefore, the minimum commuting time for the lighthouse keeper is 1.67 hours.

To find the minimum time it will take the lighthouse keeper to commute to work, we need to determine the path that minimizes the total time for both rowing and walking.

Let's consider two scenarios:

1. Rowing to the lighthouse:
The lighthouse is 2 miles offshore, and the lighthouse keeper rows at a speed of 3 miles per hour. Thus, rowing to the lighthouse will take 2 miles / 3 miles per hour = 2/3 hours = 40 minutes.

2. Walking along the coast to the lighthouse keeper's house:
The house is located 5 miles down the coast from point A. The lighthouse keeper walks at a speed of 5 miles per hour. Thus, walking along the coast will take 5 miles / 5 miles per hour = 1 hour = 60 minutes.

Therefore, the total commuting time is 40 minutes (rowing) + 60 minutes (walking) = 100 minutes.

Hence, the minimum time it will take the lighthouse keeper to commute to work is 100 minutes.