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Let's solve the problems shown in the image.

### Question 5: Determining the height of the tree
- Mary measures the angle of elevation to the top of the tree as 25° at a point 20 m from the base of the tree.

#### (a) What is the slope of the line of sight to the top of the tree?

The slope is the tangent of the angle of elevation.

\[
\text{slope} = \tan(25^\circ)
\]

Let's calculate this.

#### (b) How tall is the tree, to the nearest tenth of a meter?

The height of the tree forms the opposite side of a right triangle, with the distance from the base (20 m) as the adjacent side. Using the tangent formula:

\[
\tan(25^\circ) = \frac{\text{height}}{20}
\]

Rearrange to solve for the height (\(h\)):

\[
h = 20 \cdot \tan(25^\circ)
\]

### Question 6: Determining the angle of elevation of the sun
- Matt is 1.65 m tall, and his shadow is 1.43 m long.

#### (a) What is the slope of the angle of elevation of the sun?

The slope is the ratio of Matt's height to the length of his shadow, which is also the tangent of the angle of elevation.

\[
\text{slope} = \frac{1.65}{1.43}
\]

#### (b) Determine the angle of elevation of the sun, to the nearest degree.

To find the angle of elevation (\(\theta\)):

\[
\tan(\theta) = \frac{1.65}{1.43}
\]

\[
\theta = \tan^{-1}\left(\frac{1.65}{1.43}\right)
\]

Let me calculate these values for you.Here are the results for the problems:

### Question 5: Determining the height of the tree
- **(a)** The slope of the line of sight to the top of the tree is approximately **0.47**.
- **(b)** The height of the tree is approximately **9.3 meters**.

### Question 6: Determining the angle of elevation of the sun
- **(a)** The slope of the angle of elevation of the sun is approximately **1.15**.
- **(b)** The angle of elevation of the sun is approximately **49.1°**.

Do you need further details or have any questions?

### Relative Questions:
1. How would the slope change if the angle of elevation increased?
2. What would the height of the tree be if the angle of elevation were 30° instead of 25°?
3. How does the length of Matt's shadow affect the angle of elevation of the sun?
4. How can you calculate the angle of elevation if the tree's height and the distance from the base are known?
5. What is the relationship between the slope and the angle of elevation in right triangles?

### Tip:
Remember that the tangent function (\(\tan\)) relates the opposite and adjacent sides in a right triangle, making it very useful for solving elevation problems.

Mary wishes to determine the height of the tree in her yard. She measured the angle of elevation to the top of the tree to be 25° at a point 20 m from the base of the tree. Matt is 1.65 m tall and his shadow is 1.43 m long. What are the slopes and angles of elevation in these scenarios?

Solve problems involving the height of a tree and the angle of elevation of the sun using trigonometric ratios. Learn how to calculate slopes and angles in right triangles using tan(θ) = opposite / adjacent. Suitable for students in Grades 9-12.

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