Question
Two towns A and B are 5km and 7km, respectively, from a railroad line. The points C and D nearest to A and B on the line are 6 km apart. Where should a station be located to minimize the length of a new road from A to S to B?
Apparently the answer is supposed to be 3.5 but the methods I've used (e.g. using tan and a^2 + b^2 = c^2) but they don't seem to work. Can anyone help me get the answer? Much appreciated.
Apparently the answer is supposed to be 3.5 but the methods I've used (e.g. using tan and a^2 + b^2 = c^2) but they don't seem to work. Can anyone help me get the answer? Much appreciated.
Answers
oobleck
what is S? Does it lie on the rail line? If so then if S is x km from C, you want to minimize
z = √(x^2+5^2) + √((6-x)^2 + 7^2)
sure enough, x = 5/2, or 3.5 km away from D
dz/dx = x/√(x^2+5^2) - (6-x)/√((6-x)^2 + 7^2)
now just find x where dz/dx = 0
z = √(x^2+5^2) + √((6-x)^2 + 7^2)
sure enough, x = 5/2, or 3.5 km away from D
dz/dx = x/√(x^2+5^2) - (6-x)/√((6-x)^2 + 7^2)
now just find x where dz/dx = 0