A cowboy is 5 miles south of a stream which flows due east. He is 8 miles west and 6 miles south of his house. If he wants to allow his horse to drink from the stream, and then return to his house, how long is the shortest path he can take?

Simply go directly to his house and stop at the stream along the way, as the house is on the other side of the river. For the record, the house is about 9.4 miles away.

Originally posted by ark13 A cowboy is 5 miles south of a stream which flows due east. He is 8 miles west and 6 miles south of his house. If he wants to allow his horse to drink from the stream, and then return to his house, how long is the shortest path he can take?

Pythagorean Theorem... a²+b²=c²
a=8 b=6 ----- (8)²+(6)²=c²
64+36=c²
√100=√c²
10 = c
to determine this problem, you simply, use the Pythagorean Theorem, hence when plugging in the values for the variables, you conclude with the answer, 10 miles from the cowboys house to the stream... ðŸ™‚

Our man travels along the hypotenuse of a triangle whose other two sides are of length 5 and of length x. Then he goes along a triangle of with side lengths (8-x) and 11 (= 5 + 6). This gives a total distance s to travel of:

s = (x^2 + 5^2)^(1/2) + (11^2 + (8-x)^2)^(1/2)

The value of x which minimises s is the one for which ds/dx = 0

Of course a much easier way to do this is to reflect the cowboy to the south of the river, so that he has to go the same distance along the smaller triangle. This is a larger triangle of side lengths 8 and 11, as the guy above did. I got confused in the calculation above and has 11^2 instead of 6^2

Originally posted by DeepThought Our man travels along the hypotenuse of a triangle whose other two sides are of length 5 and of length x. Then he goes along a triangle of with side lengths (8-x) and 11 (= 5 + 6). This gives a total distance s to travel of:

s = (x^2 + 5^2)^(1/2) + (11^2 + (8-x)^2)^(1/2)

The value of x which minimises s is the one for which ds/dx = 0

ie.

x( ...[text shortened]... 6 and 2.5, clearly we want the positive one. substituting back into s above we get: 17.89 miles

Originally posted by nicklaus 10 miles exactly. Co-ordinates make a triangle and direct route to house via stream is square root of (6x6) miles + (8x8) miles =10

I should have seen that they added up to one hundred, I have no idea how I came up with that number.

Yes, you caught me out there and got me to do far more work than was neccessary - what's worse I got the wrong answer by failing to read the question properly ðŸ˜ž.