Asked by Spencer
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!
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!
Answers
Answered by
Dianne
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.
Answered by
tchrwill
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.