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A boy stands 100 meters from the bottom of a building. The angle of elevation to the roof is [tex]50^{\circ}[/tex]. The angle of elevation to the tip of the tower on the roof is [tex]60^{\circ}[/tex]. Which three statements are correct?

A. [tex]x = 52 \, \text{m}[/tex]
B. [tex]x \approx 54 \, \text{m}[/tex]
C. [tex]y = 119 \, \text{m}[/tex]
D. [tex]y \approx 117 \, \text{m}[/tex]
E. [tex]x + y \approx 173 \, \text{m}[/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to calculate the heights of the roof and tower based on the given angles of elevation.

1. Determine height \( y \) (height of the roof):

Given:
- Distance from the bottom of the building to the boy: 100 meters
- Angle of elevation to the roof: 50°

Use the tangent function, which relates an angle of a right triangle to the opposite side and adjacent side:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]

Rearranging for the opposite side (height \( y \)):
[tex]\[ y = 100 \times \tan(50^\circ) \][/tex]

Evaluating this:
[tex]\[ y \approx 100 \times 1.19175359 = 119.175359259421 \text{ meters} \][/tex]

2. Determine height \( z \) (total height to the tip of the tower):

Given:
- Angle of elevation to the tip of the tower: 60°

Using the tangent function again:
[tex]\[ \tan(\theta) = \frac{\text{opposite (total height)}}{\text{adjacent}} \][/tex]

Rearranging for the total height \( z \):
[tex]\[ z = 100 \times \tan(60^\circ) \][/tex]

Evaluating this:
[tex]\[ z \approx 100 \times 1.732050807 = 173.20508075688767 \text{ meters} \][/tex]

3. Calculate height \( x \) (height of the tower itself above the roof):

The height \( x \) is the difference between the total height \( z \) and the height of the roof \( y \):
[tex]\[ x = z - y \][/tex]

Substituting the values obtained:
[tex]\[ x \approx 173.20508075688767 - 119.175359259421 = 54.02972149746667 \text{ meters} \][/tex]

4. Sum of heights \( x \) and \( y \):

The total height \( x + y \) gives us:
[tex]\[ x + y \approx 54.02972149746667 + 119.175359259421 = 173.20508075688767 \text{ meters} \][/tex]

Upon comparing the calculated values to the statements:

Statements:
- A \( \quad x = 52 \text{ m} \) is not correct.
- B \( \quad x \approx 54 \text{ m} \) is correct.
- C \( \quad y = 119 \text{ m} \) is correct.
- D \( \quad y \approx 117 \text{ m} \) is not correct.
- E \( \quad x + y \approx 173 \text{ m} \) is correct.

The three correct statements are B, C, and E.
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