A length of rope is stretched between the top edge of a building and a stake in the ground. The head of the stake is at ground level. The rope also touches a tree that is growing halfway between the stake and the building. If the building is 40 ft tall, how tall is the tree?
(1 point)
Responses
80 ft
10 ft
20 ft
18 ft
To solve this problem, we can use the concept of similar triangles.
Let's call the height of the tree "x" and the distance from the stake to the tree "y".
The length of the rope is the sum of the distances from the tree to the stake and from the tree to the building, so it is y + x.
We can set up a proportion based on the similar triangles formed by the tree, the building, and the stake:
x/40 = (x+y)/y
Next, we can cross multiply to solve for x:
xy + x^2 = 40y
Now, let's simplify the equation:
x^2 + xy - 40y = 0
We can factor this quadratic equation:
(x - 5)(x + 8) = 0
From this equation, we can see that x = 5 or x = -8.
Since we are looking for the height of the tree, it cannot be negative, so x = 5.
Therefore, the height of the tree is 5 ft.
So, the correct answer is 5 ft.