A, B and C are 3 points on the level ground.B is due south of A . The baering of C from A is 085. A vertical mast of height 50m stands at A. The angle of elevation of T from B is 44 and the angle of elevation of T from C is 28 Calculate AB,AC, CB
the vertical mast makes a right angle with the level ground, right?
50/AC = tan28°
50/AB = tan44°
Now use the law of cosines to find BC
Hi Steve
Actually the diagram given is already drawn in 2D!. I couldnt figure out that if I think in 3 D then T is above point A and will make a right angle with the lines AB and AC
Thanks so much for opening my eyes and helping me visualize the diagram from a different angle!! If the diagram wasnt given then I think i would have noticed.
You are right.The vertical mast has to be at right angle with the level ground!
To solve this problem, we can use trigonometry and the given information about bearings, angles of elevation, and the height of the mast.
Let's start by drawing a diagram to visualize the situation. Place point A at the origin (0,0) on the level ground and draw a straight line to represent the north-south direction. Point B is due south of A, and point C is at a bearing of 085 relative to A. Place a vertical mast at A, with a height of 50m. Finally, label the angle of elevation of point T from B as 44 degrees and the angle of elevation of T from C as 28 degrees.
Now, let's solve for AB, AC, and CB step by step:
1. AB: Since B is due south of A, AB is the length of the vertical line between A and B. Given that the mast at A has a height of 50m, AB is also equal to the height of the mast. Therefore, AB = 50m.
2. AC: To find AC, we need to consider the bearings. The bearing from A to C is 085 degrees, which means that C is to the east of A.
Since the bearing is measured clockwise from the north direction, we can see that the bearing of C is 5 degrees to the east of the east direction.
Therefore, AC is the length of the horizontal line from A to C. We can calculate AC using trigonometry:
AC = AB × tan(ACB)
where ACB = 90 degrees - 5 degrees (since it forms a right angle)
= 85 degrees.
AC = 50m × tan(85 degrees).
3. CB: To find CB, we can use the angles of elevation at points B and C.
Let's draw two right triangles: one with base BT and height BT' (representing the height of T from B) and another with base CT and height CT' (representing the height of T from C).
Since the mast is vertical, BT' = CT' = 50m.
In triangle BTA:
tan(44 degrees) = BT' / BT,
BT = BT' / tan(44 degrees).
Similarly, in triangle CTA:
tan(28 degrees) = CT' / CT,
CT = CT' / tan(28 degrees).
Finally, CB = CT - BT.
To summarize:
1. AB = 50m.
2. AC = AB × tan(85 degrees).
3. BT = BT' / tan(44 degrees).
4. CT = CT' / tan(28 degrees).
5. CB = CT - BT.
Now, you can substitute the appropriate values into these equations and solve for AB, AC, and CB.
Thanks Steve,
BUt How do you know its a right angled triangle so that you can apply tan