Indirect Measurement: A tree casts a shadow 8 ft long. A 6 ft man casts a shadow 4 ft long. The triangle formed by the tree and its shadow is similar to the triangle formed by the mad and his shadow. How tall is the tree?
**PLEASE give a step-by-step explanation**
Let x = tree height
x/8 = 6/4
Solve for x.
To find the height of the tree, we can use indirect measurement and the concept of similar triangles. Let's break down the steps:
1. Understand the given information:
- The length of the tree's shadow is 8 ft.
- The length of the man's shadow is 4 ft.
2. Identify the similar triangles:
- We have two triangles formed: one by the tree and its shadow, and the other by the man and his shadow.
- The triangles are similar because they have the same shape, but not necessarily the same size.
3. Understand the properties of similar triangles:
- When two triangles are similar, their corresponding sides are proportional.
- We can set up a proportion to find the height of the tree.
4. Set up the proportion:
- Let's represent the height of the tree as 'h' and the height of the man as 'm'.
- We can write the proportion of corresponding sides as: (tree height)/(tree shadow length) = (man height)/(man shadow length).
- Substitute the given values: h / 8 = 6 / 4.
5. Solve the proportion:
- Cross-multiply: 4 * h = 6 * 8.
- Simplify: 4h = 48.
- Divide both sides of the equation by 4: h = 12.
6. Interpret the result:
- The height of the tree is 12 ft.
So, the tree is 12 feet tall.