To solve for the height of the antenna, we can break this problem into two parts:

1. Find the height of the building (from Mackenzie's eyes to the roof).
2. Find the height from Mackenzie's eyes to the top of the antenna.
3. Subtract the height of the building from the height to the top of the antenna to get the height of the antenna itself.

We can apply basic trigonometry here. Specifically, we'll use the tangent of the angle of elevation.

Step 1: Height from Mackenzie’s eyes to the roof (point A)

We can use the tangent formula:

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

Where:

• $\theta = 24^\circ$ (angle of elevation to the roof)
• $\text{adjacent} = 29 \, \text{meters}$ (horizontal distance)
• $\text{opposite} = \text{height of the building from Mackenzie’s eyes to the roof}$

Rearranging the formula to solve for the height:

$\text{height of building} = \tan(24^\circ) \times 29$

Step 2: Height from Mackenzie’s eyes to the top of the antenna (point B)

Similarly, for the angle of elevation to the top of the antenna, we have:

$\tan(32^\circ) = \frac{\text{height from Mackenzie’s eyes to the antenna}}{29}$

Rearranging to solve for the height:

$\text{height to the antenna} = \tan(32^\circ) \times 29$

Step 3: Find the height of the antenna

The height of the antenna itself is the difference between the height from Mackenzie’s eyes to the top of the antenna and the height of the building (both measured from her eyes).

$\text{height of antenna} = \text{height to the antenna} - \text{height of building}$

Now, let's compute these values.

Calculations:

1. Height of building (from Mackenzie’s eyes to the roof): $\tan(24^\circ) \times 29 = 0.4452 \times 29 = 12.91 \, \text{meters}$

2. Height to the antenna (from Mackenzie’s eyes to the top): $\tan(32^\circ) \times 29 = 0.6249 \times 29 = 18.12 \, \text{meters}$

3. Height of the antenna: $\text{height of antenna} = 18.12 - 12.91 = 5.21 \, \text{meters}$

Rounding to the nearest meter, the height of the antenna is approximately 5 meters.

Final Answer:

The height of the antenna is 5 meters.

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Would you like further clarification or additional steps explained? Here are some related questions that might help:

1. What is the relationship between the angle of elevation and the tangent function in right-angled triangles?
2. How does changing the angle of elevation affect the height calculation?
3. How can you solve for an unknown side using trigonometry in general?
4. How would the height change if Mackenzie stood further from the building?
5. What would happen if Mackenzie’s eyes were at a different height from the ground?

Tip: In problems like this, always make sure your angles are in the correct unit (degrees) and that your calculator is set to the right mode!