To solve this problem, we can use the concept of similar triangles, given the distances and heights provided.

Step 1: Convert the distances to the same unit

Since the mountain height is in feet and the distances are partly in miles, convert the distance from the base of the mountain to feet.

$1.8 \, \text{miles} = 1.8 \times 5280 = 9504 \, \text{feet}.$

Now we have:

• Distance from the base of the mountain: $9504 \, \text{feet}$
• Height of the mountain: $1449 \, \text{feet}$
• Distance from the house to the tree: $848 \, \text{feet}$

Step 2: Use similar triangles

The tree lies between the house and the mountain. Assume that the line of sight to the top of the mountain and the top of the tree forms two similar triangles. We can set up a proportion to find the height of the tree ($h$).

The ratio of the height to the base for the mountain triangle is:

$\text{Height-to-base ratio} = \frac{\text{Height of mountain}}{\text{Distance to base of mountain}} = \frac{1449}{9504}.$

The height of the tree ($h$) corresponds to the shorter triangle formed by the house and the tree. The base of this smaller triangle is $848 \, \text{feet}$.

$\frac{h}{848} = \frac{1449}{9504}.$

Step 3: Solve for $h$

Cross-multiply to solve for $h$:

$h = \frac{1449 \times 848}{9504}.$

Calculate:

$h = \frac{1227552}{9504} \approx 129 \, \text{feet}.$

Final Answer:

The height of the tree is approximately 129 feet.

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Would you like a detailed explanation of how similar triangles apply in this case, or do you have any further questions?

Related Questions:

1. What is the importance of similar triangles in solving real-world problems?
2. How can you calculate distances or heights when only angles are given?
3. What are other ways to measure the height of an object like a tree or a building?
4. Can we use trigonometry to solve this problem differently? If so, how?
5. How does this calculation change if the distances were not measured along flat ground?

Tip:

When working with real-world measurements, always ensure all units are consistent before setting up proportions or equations.