Problema Solution
A tree is supported by ropes. One rope goes from the top of the tree to a point on the ground. The height of the tree is 4 feet more than twice the distance between the base of the tree and the rope anchored in the ground. The length of the rope is 6 feet more than twice the distance between the base of the tree and the rope anchored in the ground. Find the height of the tree.
Answer provided by our tutors
let
l = the length of the rope, l>0
h = the height of the tree, h>0
d = the distance between the base of the tree and the rope anchored in the ground, d>0
The height of the tree is 4 feet more than twice the distance between the base of the tree and the rope anchored in the ground.
h = 4 + 2d
The length of the rope is 6 feet more than twice the distance between the base of the tree and the rope anchored in the ground.
l = 6 + 2d
Using the Pythagorean Theorem we have:
l^2 = d^2 + h^2
plug h = 4 + 2d and l = 6 + 2d into the last equation:
(6 + 2d)^2 = d^2 + (4 + 2d)^2
by solving we find:
d = 10 ft
click here to see the step by step solution of the quadratic equation:
h = 4 + 2*10
h = 24 ft
the tree is 24 feet high.