There are two vertical poles, one of height 100 feet and the other 70 feet, positioned a horizontal distance of 80 feet apart on level ground. A rope of length 100 feet connects the tops of the two poles. A weight, placed on the rope so that it can slide freely along it, is allowed to come to rest.
Find the horizontal distance of the weight from the 100 feet pole and the vertical distance from the ground.
The trick is to re-imagine the problem slightly as a large right triangle with no inflection point.
Call the first pillar AB, with height 70, and A at{0,70}, and the second CD, with height 100, and C at {80,100}
A circle of size 10 on C will then intersect the y-axis at {0,4} and {0,16}. We can ignore the latter for current purposes.
Connect C and {0.4} with a segment, CD. To be sure, this line doesn't have an inflection point, but we know by reason of similar triangles and reflection that the inflection will take place on a line parallel to the x-axis and passing exactly halfway between D and A, i.e. at {0,55}. Construct this line. The intersection of the line and the line CD gives the desired coordinates at {20,55}, giving the same result as Brian's.
Geometrically, with heights a and b, a<b, a horizontal distance c, and rope r, and using the same construction, we have c^2+(b-y)^2= r^2), giving y and so D, and then (1/2)*(a-y) for the line, with x following automatically, since we already know the y-coordinate and the slope of CD.
Edited on June 16, 2019, 8:40 pm
|
|
Posted by broll
on 2019-06-16 04:23:51 |
A nice trick question
