Kim measured the angle between the base of the tree and the top of the tree and received a reading of 1.2 degrees. If Kim was 1500 feet from the base of the tree when the measurement was made, how high was the tree?

To find the height of the tree, we can use basic trigonometry, specifically the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side (the height of the tree) to the adjacent side (the distance from the base of the tree).

The formula is:

\[
\tan(\theta) = \frac{\text{height of the tree}}{\text{distance from the tree}}
\]

Where:
- \(\theta = 1.2^\circ\) (the angle Kim measured)
- Distance = 1500 feet (Kim's distance from the tree)

We need to solve for the height of the tree:

\[
\text{height of the tree} = \tan(1.2^\circ) \times 1500
\]

Now, calculating the tangent of 1.2 degrees:

\[
\tan(1.2^\circ) \approx 0.02094
\]

Thus,

\[
\text{height of the tree} = 0.02094 \times 1500 \approx 31.41 \, \text{feet}
\]

So, the height of the tree is approximately **31.41 feet**.

Would you like more details on this, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. What is the difference between sine, cosine, and tangent functions?
2. How does changing the angle impact the height of the tree in this scenario?
3. How would the height calculation change if Kim was closer or farther from the tree?
4. What if Kim measured the angle from a higher elevation, like a hill? How would that change the approach?
5. How can trigonometry be used in other real-life applications, like measuring the height of buildings?

**Tip**: For small angles, the tangent function gives very small values, meaning slight changes in angle can lead to significant changes in height estimates.

Learn how to use trigonometry to calculate the height of a tree when the angle to the top of the tree is 1.2 degrees and the distance from the base is 1500 feet. This problem covers key concepts such as the tangent function and basic trigonometric formulas. Suitable for high school students (Grades 9-12).

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