Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
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.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.