Two points R and S are on level ground 350m apart. The bearing of S from R is 147°. From R, the bearing of a radio mast is 055°, and from S, the bearing is 032°. The angle of elevation of the top of the mast from R is 20°. Find the height of the mast
Draw a diagram. Let
M = base of mast
T = top of mast
In ∆RSM, the angles are
R=85°, S=72°, so M=23°
using the law of sines,
s/sinS = m/simM. That is,
RM/sin72° = 350/sin23°
RM = 851.91
To find the mast's height MT, then, solve
MT/RM = tan20°
To find the height of the mast, we can use the concept of trigonometry.
Step 1: Draw a diagram to represent the given information.
- Place points R and S on level ground, 350m apart.
- Measure the bearing of S from R, which is 147°.
- Measure the bearing of the radio mast from R, which is 055°.
- Measure the bearing of the radio mast from S, which is 032°.
- Draw a line representing the angle of elevation from R to the top of the mast, with an angle of 20°.
Step 2: Determine the angles of the triangle formed by R, S, and the top of the mast.
- Label the angle between R, S, and the top of the mast as angle RSM (θ1).
- Label the angle between R, the top of the mast, and the horizontal line as angle MRH (θ2).
- Label the angle between S, the top of the mast, and the horizontal line as angle MSH (θ3).
Step 3: Find the value of θ1.
- Since θ1 is the interior angle of a triangle, θ1 = 180° - 147° = 33°.
Step 4: Find the values of θ2 and θ3.
- Since the bearings are measured from the North in a clockwise direction, we can calculate:
- θ2 = 180° - 55° = 125°
- θ3 = 360° - 32° = 328°
Step 5: Use the trigonometric ratio to find the height of the mast.
- In triangle RSM, we have the opposite side (height of the mast) and the adjacent side.
- We can use the tangent ratio: tan(θ1) = opposite/adjacent
- tan(θ1) = height of the mast/350m
Step 6: Solve for the height of the mast.
- height of the mast = tan(θ1) * 350m
Finally, plug in the values and calculate the height of the mast.
height of the mast = tan(33°) * 350m = 0.6494 * 350m = 227.39m
Therefore, the height of the mast is approximately 227.39 meters.
To find the height of the mast, we can use trigonometry and the given information.
First, let's draw a diagram to visualize the problem. Draw two points R and S on level ground, 350m apart. Label the angle between the line connecting R and S and the horizontal line as A, and label the angle between the line connecting R and the top of the mast and the horizontal line as B.
From the given information, we know that the bearing of S from R is 147°, so angle A is 147°. The bearing of the mast from R is 55°, so angle B is 55°. The angle of elevation of the top of the mast from R is 20°, so angle C is 20°.
Now, let's break down the problem into two triangles, triangle RAS and triangle RBT, where T is the top of the mast.
In triangle RAS, we have two sides and an included angle. We can use the Law of Cosines to calculate the length of RS:
RS^2 = RA^2 + SA^2 - 2 * RA * SA * cos(A)
Since RA = SA (they are both equal to 350m), we can simplify the equation to:
RS^2 = 2 * RA^2 * (1 - cos(A))
Now, let's calculate RS:
RS^2 = 2 * (350m)^2 * (1 - cos(147°))
RS^2 ≈ 2 * (122500m^2) * (1 - (-0.6428))
RS^2 ≈ 2 * (122500m^2) * (1 + 0.6428)
RS^2 ≈ 2 * (122500m^2) * (1.6428)
RS^2 ≈ 40302500m^2
RS ≈ √40302500m^2
RS ≈ 6351.88m
Now, let's focus on triangle RBT. We want to find the height of the mast, which is TB. We can use the tangent function to find TB:
tan(B) = TB / RB
tan(55°) = TB / RB
tan(55°) = TB / 350m
TB ≈ tan(55°) * 350m
TB ≈ 1.42815 * 350m
TB ≈ 499.35m
So, the height of the mast, TB, is approximately 499.35 meters.