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 36 feet tall, how tall is the tree?
A rope stretched between a building and a stake in the ground, with a tree growing halfway between the building and the stake and touching the rope.
The building is at the left of the image.
A point labeled 'stake' is at the right of the image on the same horizontal line as the bottom of the building.
A tree is drawn in between the building and the stake.
A line segment is drawn connecting the stake to the bottom of the tree and the bottom of the building. A single tick mark is drawn on the part of the segment between the bottom of the tree and the bottom of the building. An equal single tick mark is drawn on the part of the segment between the bottom of the tree and the stake.
A line segment is drawn connecting the stake to the top of the tree and the top of the building. Double tick marks are drawn on the part of the segment between the top of the tree and the top of the building. Equal double tick marks are drawn on the part of the segment between the top of the tree and the stake.
(1 point)
Responses
36 ft.
36 ft.
18 ft.
18 ft.
9 ft.
9 ft.
27 ft.
The tree is 18 feet tall.
To solve this problem, we can use the concept of similar triangles. Let's call the height of the tree "h".
According to the problem, the rope touches the tree, the building, and the stake. This means that the rope forms a right triangle with the tree, the building, and the stake.
Since the rope is stretched between the top edge of the building and the stake, the length of the rope is equal to the height of the building plus the height of the tree (h + 36).
We also know that the tree is growing halfway between the stake and the building. This means that the distance from the stake to the tree is equal to the distance from the tree to the building.
Therefore, we can set up a proportion: (height of tree) / (distance from stake to tree) = (height of building) / (distance from tree to building).
Using the given values, we have: h / (h/2) = 36 / (h/2)
Simplifying the equation, we get: 2h / (h/2) = 36 / (h/2)
Multiplying both sides of the equation by (h/2), we get: 2h = 36
Dividing both sides of the equation by 2, we find: h = 18
Therefore, the height of the tree is 18 feet.
To find the height of the tree, we can use the concept of similar triangles. Let's name the height of the tree "h".
Since the tree is growing halfway between the stake and the building, the distance from the stake to the tree is also halfway between the distance from the stake to the building. Let's call this distance "d".
Now, we can form two similar triangles: one with the building, the tree, and the stake, and another with the building, the rope, and the stake.
In the first triangle, the ratio of the height of the tree to the distance from the stake to the tree is the same as the ratio of the height of the building to the distance from the stake to the building. Since the height of the building is given as 36 feet and the distance from the stake to the building is "d", we have:
h / d = 36 / d
In the second triangle, the ratio of the length of the rope to the distance from the stake to the rope is the same as the ratio of the height of the building to the distance from the stake to the building. Since the length of the rope is equal to the height of the building (given as 36 feet), we have:
36 / d = 36 / d
Comparing the two equations, we can see that they are equal. Therefore, we can conclude that:
h / d = 36 / d
By canceling out the "d" terms on both sides, we are left with:
h = 36
So, the height of the tree is 36 feet.