**The angle of elevation of the top of a lamp post 8 m tall, from a point on the ground is 45**o**. Find the distance of the point from the foot of the lamp post.**

To solve this problem, we can use trigonometry, specifically the tangent function, which relates the angle of elevation to the opposite side and adjacent side in a right triangle.

### Given:
- Height of the lamp post \( h = 8 \) meters.
- Angle of elevation \( \theta = 45^\circ \).
- Distance from the point on the ground to the foot of the lamp post \( d \) (what we need to find).

### Step-by-Step Solution:

1. **Set up the tangent function:**

   The tangent of an angle in a right triangle is given by:

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

   Here, the opposite side is the height of the lamp post, and the adjacent side is the distance from the point to the foot of the lamp post.

   \[
   \tan(45^\circ) = \frac{h}{d}
   \]

2. **Substitute the given values:**

   \[
   \tan(45^\circ) = \frac{8}{d}
   \]

3. **Simplify:**

   Since \( \tan(45^\circ) = 1 \), we have:

   \[
   1 = \frac{8}{d}
   \]

4. **Solve for \( d \):**

   Multiply both sides by \( d \):

   \[
   d = 8 \text{ meters}
   \]

### Final Answer:
The distance of the point from the foot of the lamp post is \( 8 \) meters.

Would you like more details or have any questions?

Here are 5 related questions:
1. What would happen to the distance if the angle of elevation were decreased?
2. How would you solve the problem if the lamp post were taller, say 10 meters?
3. What trigonometric function would you use if the angle of elevation were 30° instead of 45°?
4. How can you determine the height of the lamp post if the distance is known?
5. What if the angle of elevation were 60°? How would the distance change?

**Tip:** Always remember that a 45° angle in a right triangle indicates that the opposite and adjacent sides are equal in length.

Learn how to find the distance from a point on the ground to the foot of an 8m tall lamp post using trigonometry. Step-by-step solution provided with details on using the tangent function. Suitable for high school students.

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