This problem involves using similar triangles to estimate the height of a tree. Based on the diagram and the given dataThis problem involves using similar triangles to estimate the height of a tree. Based on the diagram and the given

• The tree casts a shadow of 10 ft.
• A person with a height of 6 ft casts a shadow of 4 ft.

We can set up a proportion using the fact that the triangles formed by the tree and its shadow, and the person and their shadow, are similar. This means the ratios of corresponding sides are equal:

$\frac{\text{Height of Tree}}{\text{Shadow of Tree}} = \frac{\text{Height of Person}}{\text{Shadow of Person}}$

Let's assign the values:

• Let the height of the tree be $h$ (unknown).
• Shadow of the tree = 10 ft.
• Height of the person = 6 ft.
• Shadow of the person = 4 ft.

Now set up the proportion:

$\frac{h}{10} = \frac{6}{4}$

Solve for $h$:

$h = \frac{6}{4} \times 10 = 1.5 \times 10 = 15 \, \text{ft}$

So, the estimated height of the tree is 15 feet.

Would you like any more details or clarifications on this solution?

Here are 5 related questions for further practice:

1. If the person's height was 5 ft and their shadow was 3.5 ft, how would the height of the tree change?
2. What would happen if the tree’s shadow was 12 ft instead of 10 ft, while keeping the other values the same?
3. How would you solve the problem if the shadow measurements were in meters instead of feet?
4. What is the relationship between the height of an object and the length of its shadow in similar triangle problems?
5. How would you calculate the height of the tree if the height of the person wasn’t given?

Tip: Similar triangles are very useful in real-life situations where direct measurement of heights (like that of a tall tree) is not feasible!