I've been working forever to figure out this problem!
If two flagpoles are 10m and 70m tall and are 100m apart, find the height of the point where a line from the top of the first to the bottom of the second intersects a line from the bottom of the first to the top of the second.
I've drawn a picture but I can't do it after that! Please help!
2 answers
This one is hard to answer without showing you a picture but here goes. The line from the top of the 10m pole to the base of the 70m pole is the hypotenuse of a right triangle. Dropping a vertical line from the point of intersection forms a second triangle on the end that is similar to the original and the sides are proportional. Label the vertical line z and the horizontal line x. You can make a proportion now from the two triangles. 10/100 = z/x. In a similar way a proportion can be made with the 70m pole. The vertical would still be z but the horizontal could be y which will make the proportion 70/100 = z/y. Cross multiplying the proportions gives 10x=100z and 70y=100z. Since both 10x and 70y are equal to 100z they are equal to each other BUT y = 100-x. With substitution you get 10x=70(100-x). Solve this equation for x then substitute the answer in the very first equation to find z. z is the height you want.
If two flagpoles are 10m and 70m tall and are 100m apart, find the height of the point where a line from the top of the first to the bottom of the second intersects a line from the bottom of the first to the top of the second.
The historical basis of the problem:
Two ladders of different lengths are leaning against two buildings with their bases against the opposite building.
Given: The heights where the ladders touch the buildings, A and B.
Find: The height of the point where they cross, X.
Assume the following picture:
I*
I.....*....................................................*{
I...........*.......................................*...... I
I..................*.........................*..............I
IA......................*............*.....................I
I..............................*............................{B
I........................*.....{.....*......................I
I................*............ I X.........*...............{
I........*.....................I...................*........{
I*________________ I______________ * I
(C - y) C Y
1--Let A and B = the two height of the ladders against the buildings.
2--Let X = the height of the ladder crossing.
3--From the figure, A/C = X/Y or AY = CX.
4--Similarly, B/C = X/(C - Y) or BY = BC - CX.
5--Y = CX/A = (BC - CX)/B from which X = AB/(A+B).
Note - X is actually one half the harmonic mean of the two dimensions A and B, the harmonic mean being 2AB/(A + B).
Therefore, the height of the crossing is totally independant of the distance between the two buildings.
The historical basis of the problem:
Two ladders of different lengths are leaning against two buildings with their bases against the opposite building.
Given: The heights where the ladders touch the buildings, A and B.
Find: The height of the point where they cross, X.
Assume the following picture:
I*
I.....*....................................................*{
I...........*.......................................*...... I
I..................*.........................*..............I
IA......................*............*.....................I
I..............................*............................{B
I........................*.....{.....*......................I
I................*............ I X.........*...............{
I........*.....................I...................*........{
I*________________ I______________ * I
(C - y) C Y
1--Let A and B = the two height of the ladders against the buildings.
2--Let X = the height of the ladder crossing.
3--From the figure, A/C = X/Y or AY = CX.
4--Similarly, B/C = X/(C - Y) or BY = BC - CX.
5--Y = CX/A = (BC - CX)/B from which X = AB/(A+B).
Note - X is actually one half the harmonic mean of the two dimensions A and B, the harmonic mean being 2AB/(A + B).
Therefore, the height of the crossing is totally independant of the distance between the two buildings.
2 answers