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?
Since the rope stretches from the top edge of the building to the stake in the ground and touches the tree halfway between them, this forms a right triangle.
Let the height of the tree be "x" ft.
The distance from the stake to the tree is "x" ft.
The distance from the tree to the building is also "x" ft.
The total height of the rope is the height of the building + the height of the tree.
Thus, the total height of the rope is 40 ft + x ft = 40 + x ft.
By the Pythagorean theorem, we have the following equation: (40 + x)^2 = 40^2 + x^2.
Expanding the left side gives 1600 + 80x + x^2 = 1600 + x^2.
Subtracting x^2 from both sides gives 80x = 0.
Dividing both sides by 80 gives x = 0. ☐
half as tall as the building (similar triangle)
Let's assume the height of tree is 'h' feet.
Since the tree is growing halfway between the stake and the building, the distance from the stake to the tree is the same as the distance from the tree to the building. Assuming this distance is 'd' feet, we have:
d + d = 40
2d = 40
d = 20 feet
Now we have a similar triangle formed by the height of the building, the height of the tree, and the distance from the tree to the building.
According to the property of similar triangles, the ratio of corresponding sides is equal. Therefore, we can set up the following proportion:
h/40 = (h/2)/20
Cross multiplying, we get:
20h = 40 * (h/2)
20h = 20h
So, the height of the tree is half the height of the building i.e., h = 40/2 = 20 feet. Answer: \boxed{20}.
To determine the height of the tree, we need to find the distance from the ground to the point where the rope touches the tree.
Let's denote the height of the tree as 'h'.
Since the rope is stretched from the top edge of the building to the stake in the ground, the total length of the rope is also the hypotenuse of a right-angled triangle formed by the building, the tree, and the rope.
So, by using the Pythagorean theorem, we can solve for 'h':
(Length of rope)^2 = (Height of the building)^2 + (Distance from stake to tree)^2
The length of the rope is the sum of the height of the building (40 ft) and the height of the tree (h). The distance from the stake to the tree is half of the distance between the stake and the building, which is 40 ft/2 = 20 ft.
Plugging in these values into the equation:
(40 + h)^2 = 40^2 + 20^2
Simplifying:
1600 + 80h + h^2 = 1600 + 400
Combining like terms:
h^2 + 80h = 400
Rearranging the equation to isolate 'h':
h^2 + 80h - 400 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
Factoring it would give us:
(h + 100)(h - 4) = 0
Setting each factor equal to zero:
h + 100 = 0 or h - 4 = 0
Solving for 'h':
h = -100 or h = 4
Since the height of an object cannot be negative, we discard -100 ft as a solution. Therefore, the height of the tree is 4 ft.