Question

Find the height of station B.

Answer 1

Step 1: Understanding the problem:
We are asked to find the height of station B. Based on the given passage, we know the following:
- The distance between the base C of the tower and point O is 36 m.
- The angle of elevation of the top of station B from point O is 30º.
- We need to find the height of station B, which we can represent as \( h_B \).

Step 2: Using trigonometry to find the height:
We can use the tangent function, which relates the angle of elevation, the height of the tower, and the horizontal distance between the point of observation and the base of the tower.
The formula for the tangent of an angle \( \theta \) in a right triangle is:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \] In this case, the opposite side is the height of station B, \( h_B \), and the adjacent side is the distance from point O to the base C of the tower, which is 36 m. For station B, the angle of elevation is 30º.
Thus, the equation becomes:
\[ \tan 30^\circ = \frac{h_B}{36} \] We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \). Substituting this into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h_B}{36} \]

Step 3: Solving for \( h_B \):
To find \( h_B \), multiply both sides of the equation by 36:
\[ h_B = 36 \times \frac{1}{\sqrt{3}} = \frac{36}{\sqrt{3}} \] Now, rationalize the denominator by multiplying both the numerator and denominator by \( \sqrt{3} \):
\[ h_B = \frac{36 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{36 \sqrt{3}}{3} = 12\sqrt{3} \]

Step 4: Conclusion:
The height of station B is \( 12\sqrt{3} \) meters.

Answer 2

Step 1: Understanding the problem:
We are asked to find the height of station A. Based on the given passage, we know the following:
- The distance between the base C of the tower and point O is 36 m.
- The angle of elevation of the top of station A from point O is 45º.
- We need to find the height of station A, which we can represent as \( h_A \).

Step 2: Using trigonometry to find the height:
We can use the tangent function, which relates the angle of elevation, the height of the tower, and the horizontal distance between the point of observation and the base of the tower.
The formula for the tangent of an angle \( \theta \) in a right triangle is:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \] In this case, the opposite side is the height of station A, \( h_A \), and the adjacent side is the distance from point O to the base C of the tower, which is 36 m. For station A, the angle of elevation is 45º.
Thus, the equation becomes:
\[ \tan 45^\circ = \frac{h_A}{36} \] We know that \( \tan 45^\circ = 1 \). Substituting this into the equation:
\[ 1 = \frac{h_A}{36} \]

Step 3: Solving for \( h_A \):
To find \( h_A \), multiply both sides of the equation by 36:
\[ h_A = 36 \times 1 = 36 \]

Step 4: Conclusion:
The height of station A is 36 meters.

Answer 3

Step 1: Understanding the problem:
We are asked to find the lengths of the wires \( OA \) and \( OB \) that support the radio tower.
The distance between the base C of the tower and point O is 36 m, and the angles of elevation for station A and station B are 45º and 30º, respectively.

Step 2: Using trigonometry to find the length of wire OA:
We know that the angle of elevation to station A is 45º, and the height of station A is 36 m. We can use the Pythagorean theorem to find the length of the wire \( OA \), which is the hypotenuse of a right triangle.
In a right triangle, the relationship between the sides is given by the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2 \] For station A, the base is the distance from point O to the base C of the tower (36 m), and the height is the height of station A (36 m). Therefore, the length of wire OA, denoted as \( OA \), is given by:
\[ OA^2 = 36^2 + 36^2 = 1296 + 1296 = 2592 \] Now, take the square root of both sides to find \( OA \):
\[ OA = \sqrt{2592} = 36\sqrt{2} \]

Step 3: Using trigonometry to find the length of wire OB:
For station B, the angle of elevation is 30º, and the height of station B is \( 12\sqrt{3} \) meters (from the previous calculation). We can again use the Pythagorean theorem to find the length of the wire \( OB \).
The height of station B is \( 12\sqrt{3} \) m, and the base is the distance from point O to the base C of the tower (36 m). Therefore, the length of wire OB, denoted as \( OB \), is given by:
\[ OB^2 = 36^2 + (12\sqrt{3})^2 = 1296 + 432 = 1728 \] Now, take the square root of both sides to find \( OB \):
\[ OB = \sqrt{1728} = 24\sqrt{3} \]

Step 4: Conclusion:
The length of wire OA is \( 36\sqrt{2} \) meters, and the length of wire OB is \( 24\sqrt{3} \) meters.