math puzzle

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.

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

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?

Sorry, I intended to say that the he was 6 miles north of his house, so he's between the two.

In that case the shortest distance would be 18.6015 miles.

5 miles north to the river and then along the hyp. of the 8,11,13.6015 triangle.

If he used the hyp. of the smaller triangle to reach the river and then went due south the 11 miles to his house it would be 20.4340 miles.

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(x^2 + 25)^(-1/2) - (8-x)(121 + (8-x)^2)^(-1/2) = 0

mulitply out the fractions and square to give:

x^2 (185 - 16x +x^2) = (8 - x)^2 (x^2 + 25)

expand and simplify to get:

96x^2 + 400x - 1600 = 0

which has roots: -1.6 and 2.5, clearly we want the positive one. substituting back into s above we get: 17.89 miles

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

Originally posted by ark13
Yup, very nice. 😀