To solve this, we can use trigonometry. Specifically, the tangent function, which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (distance from the base of the tower).

Given:

• Distance from the base of the tower (adjacent side): $400 \, \text{m}$
• Angle of elevation: $46^\circ$

We need to find the height of the tower (opposite side).

Formula:

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

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

We can rearrange this to solve for the height of the tower:

$\text{height} = \tan(\theta) \times \text{adjacent}$

Substituting the known values:

$\text{height} = \tan(46^\circ) \times 400$

Now, calculating:

$\text{height} \approx 0.4295 \times 400 \approx 171.8 \, \text{m}$

Final Answer:

The height of the tower is approximately 171.8 meters.

Would you like to explore any further details or have any questions about the process?

Related questions:

1. How would the height change if the angle of elevation was different?
2. How can this problem be solved using sine or cosine functions?
3. How do you find the distance to the top of the tower (hypotenuse)?
4. How would you solve this if the hiker was at a different distance?
5. What is the formula for calculating the height if you know the angle of depression?

Tip:

Remember, the tangent function is useful for solving right triangles when you know the angle of elevation (or depression) and one of the sides.