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The problem visible in the image reads as follows:

**23. From the top of a watchtower that is 14 feet high, a forest ranger finds that the angle of depression of a campfire on the ground below is 28°40′. If the bottom of the tower is in the same horizontal plane as the campfire, how far is the bottom of the tower from the campfire?**

This is a typical trigonometry problem involving the angle of depression and a right triangle.

### Solution:

1. Let the height of the watchtower be \( h = 14 \) feet.
2. The angle of depression is \( 28^\circ 40' \).
3. We need to find the horizontal distance \( d \) from the base of the tower to the campfire.

The angle of depression from the top of the tower is equal to the angle of elevation from the campfire to the top of the tower, which is also \( 28^\circ 40' \). We can use the tangent function since it relates the opposite side (the height of the tower) and the adjacent side (the distance from the base of the tower to the campfire):

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\]

So,

\[
\tan(28^\circ 40') = \frac{14}{d}
\]

Rearranging for \( d \):

\[
d = \frac{14}{\tan(28^\circ 40')}
\]

Let me calculate the exact value.The horizontal distance from the base of the tower to the campfire is approximately 25.61 feet.

Would you like more details on this solution or have any further questions?

Here are 5 related questions for further practice:
1. If the height of the tower were 20 feet and the angle of depression was 30°, how far would the campfire be?
2. How would the result change if the angle of depression were 45° with the same 14-foot tower?
3. Can you calculate the angle of depression if the distance from the base of the tower to the campfire was 30 feet?
4. If the height of the watchtower were doubled, how would the distance to the campfire change?
5. How is the tangent function used to solve right triangle problems involving angles of depression?

**Tip:** Always ensure your angle units (degrees or radians) match the functions you're using in your calculator!

From the top of a watchtower that is 14 feet high, a forest ranger finds that the angle of depression of a campfire on the ground below is 28°40′. If the bottom of the tower is in the same horizontal plane as the campfire, how far is the bottom of the tower from the campfire?

Find out how to calculate the distance from a watchtower to a campfire using trigonometric concepts. Learn how the tangent function helps solve problems involving angles of depression and right triangles. This problem involves a 14-foot high tower and a 28°40′ angle of depression.

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