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.
