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 tree is 12 feet tall, how tall is the building?
32 ft
8ft
24ft
16ft
24ft
To find the height of the building, we can use the concept of similar triangles. Let's call the height of the building 'x' feet.
We know that the rope touches the top of the building, the stake in the ground, and the middle of the tree. This forms two triangles: one formed by the tree and the stake, and another formed by the tree and the building.
Since the tree is halfway between the stake and the building, the height of the tree is half the height of the building. So, the height of the tree is 12 feet, which means that the height of the building is 2 times the height of the tree, or 2 * 12 = 24 feet.
Therefore, the correct answer is 24 ft.
To find the height of the building, we can use similar triangles. Let's denote the height of the building as 'h' and the distance between the stake and the building as 'd'.
Since the rope touches the tree that is halfway between the stake and the building, we can assume that the distance between the stake and the tree is also 'd'.
Using the Pythagorean theorem, we can set up the following equation:
(d)^2 + (12)^2 = (d+h)^2
Expanding and simplifying the equation, we get:
d^2 + 144 = d^2 + 2dh + h^2
The d^2 cancels out on both sides of the equation, leaving us with:
144 = 2dh + h^2
Since the tree is halfway between the stake and the building, we know that dh = h/2. Substituting this into the equation, we get:
144 = h^2/2 + h^2
Multiplying both sides of the equation by 2, we get:
288 = h^2 + 2h^2
Combining like terms, we have:
288 = 3h^2
Dividing both sides of the equation by 3, we get:
h^2 = 96
Taking the square root of both sides, we get:
h = √96
Simplifying, we have:
h = 4√6
So, the height of the building is approximately 4√6 feet.