A person who is 6
feet tall is standing 120
feet from the base of a tree, and the tree casts a 130
foot shadow. The person's shadow is 10
feet in length. What is the height of the tree?
Let's use a proportion to solve the problem:
(tree height) / (tree shadow length) = (person height) / (person shadow length)
Plugging in the given values, we get:
(tree height) / 130 = 6 / 10
Simplifying the right side, we get:
(tree height) / 130 = 3 / 5
Cross-multiplying, we get:
5(tree height) = 130(3)
5(tree height) = 390
Dividing both sides by 5, we get:
tree height = 78
Therefore, the height of the tree is 78 feet.
To find the height of the tree, we can use similar triangles.
We have two similar triangles: one formed by the person, the person's shadow, and a portion of the tree, and the other formed by the entire tree and its shadow.
Let's label the height of the person as "x" and the height of the tree as "h".
Using the properties of similar triangles, we can set up the following proportion:
(x / 10) = (h / 130)
To solve for "h", we can cross-multiply and solve for "h":
x * 130 = 10 * h
130x = 10h
Divide both sides by 10:
13x = h
Therefore, the height of the tree, "h", is equal to 13 times the height of the person, "x".