C and D are two observation posts on the same horizontal ground at the port A of a vertical lower AB . The lower is 18m due north of D and 24m east of C. The angle of elevation of B from D 135 . Calculate to three significant figures:
1) the height AB
2) distance
3) the angle of elevation of B from C
4) bearing of C from B
To solve this problem, we can use trigonometry and the given information.
1) To find the height AB, we can use the tangent function. Let's call the height AB as h.
In triangle ABD, the angle of elevation of B from D is 135 degrees. We can use the tangent function:
tan(135) = h/18
h = 18 * tan(135)
h ≈ 18 * (-1)
h ≈ -18 meters
Therefore, the height AB is approximately -18 meters.
2) To find the distance between C and D, we can use the Pythagorean theorem. Let's call the distance between C and D as x.
Using the given information, we know that C is 24 meters east of D. So, the horizontal distance CD is 24 meters.
In triangle CD, the hypotenuse is CD and one of the legs is x. The other leg is the horizontal distance 24 meters.
Applying the Pythagorean theorem:
x^2 + 24^2 = CD^2
Since CD is the hypotenuse, CD^2 = 24^2 + x^2.
CD^2 = 576 + x^2
Since AB is the hypotenuse of the triangle ABD, AB = CD.
Therefore, AB^2 = 576 + x^2.
3) To find the angle of elevation of B from C, we can use the tangent function. Let's call the angle of elevation of B from C as θ.
In triangle BCA, we know that the height AB is -18 meters and the horizontal distance CB is 24 meters.
Using the tangent function:
tan(θ) = (-18)/24
θ = atan(-18/24)
θ ≈ -37.87 degrees
Therefore, the angle of elevation of B from C is approximately -37.87 degrees.
4) To find the bearing of C from B, we can use the concept of bearings. Bearings are measured clockwise from the north direction. Let's call the bearing of C from B as α.
The bearing of C from B is the angle between the line BC and the north direction.
Since C is east of B, the bearing of C from B can be found by subtracting the angle of elevation of B from C from 90 degrees.
α = 90 - (-37.87)
α ≈ 127.87 degrees
Therefore, the bearing of C from B is approximately 127.87 degrees.