We will calculate the height of the building by using trigonometry. Here's the breakdown of the problem:

1. Known Information:

   • Distance from point $P$ to the base of the cliff = $543 \, \text{ft}$.
   • Angle of elevation to the top of the cliff = $56.8^\circ$.
   • Angle of elevation to the top of the building = $72.4^\circ$.

2. Unknowns:

   • Height of the cliff ($h_c$).
   • Height of the building above the cliff ($h_b$).
   • Total height of the building ($h_{\text{total}} = h_c + h_b$).

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Step 1: Find the height of the cliff ($h_c$)

Using the tangent function: $\tan(56.8^\circ) = \frac{h_c}{543}$ Rearrange to solve for $h_c$: $h_c = 543 \cdot \tan(56.8^\circ)$

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Step 2: Find the height to the top of the building ($h_{\text{total}}$)

Using the tangent function again: $\tan(72.4^\circ) = \frac{h_{\text{total}}}{543}$ Rearrange to solve for $h_{\text{total}}$: $h_{\text{total}} = 543 \cdot \tan(72.4^\circ)$

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Step 3: Find the height of the building ($h_b$)

$h_b = h_{\text{total}} - h_c$

Let’s calculate. math

Given values

distance = 543 # ft angle_cliff = math.radians(56.8) # Convert to radians angle_building = math.radians(72.4) # Convert to radians

Height of the cliff

h_cliff = distance * math.tan(angle_cliff)

Total height (cliff + building)

h_total = distance * math.tan(angle_building)

Height of the building

h_building = h_total - h_cliff

h_cliff, h_total, h_building### Results:

1. Height of the cliff ($h_c$): $830 \, \text{ft}$ (rounded to the nearest foot).
2. Total height to the top of the building ($h_{\text{total}}$): $1712 \, \text{ft}$ (rounded to the nearest foot).
3. Height of the building ($h_b$): $882 \, \text{ft}$ (rounded to the nearest foot).

Thus, the height of the building is approximately $882 \, \text{ft}$.

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Would you like me to explain any step in more detail?
Here are 5 questions to deepen understanding:

1. What does the tangent function represent in this context?
2. How does the distance between point $P$ and the cliff affect the calculations?
3. What other methods could we use to calculate the building's height?
4. How would the problem change if the angles were measured in radians instead of degrees?
5. What practical applications might this method of calculation have in real life?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems.