- 23 Mar '07 22:55

Pythagorean Theorem... a²+b²=c²*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?**

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... - 24 Mar '07 01:13Our 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 - 24 Mar '07 10:13 / 2 editsOf 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
- 24 Mar '07 20:15

Yup, very nice.*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