A surveyor wishes to find the height of a mountain. He selects points A and B on level ground along the same line of sight to the mountain. He measures the distance AB to be 1503 m and the angles of elevation of the top of the mountain from the points to be 16.9° from A and 22.3° from B.
a) Calculate the distance to the top of the mountain from point B.
The triangle that A and B and their angles of elevation makes, I called triangle ABC; I calculated C's angle to be 180°-16.9°-22.3°=140.8°... I feel like this is where I went wrong, is it incorrect to do this? I probably visualised it incorrectly again or something..
Anyway, my solution, using sine law:
b/sin22.3° = 1503/140.8°
b = 1503sin22.3°/140.8°
b = 4
The answer should be 6060m. So I'm clearly doing something completely wrong here.
Sorry, a) should say, "calculate the distance to the top of the mountain from point A," not point B.
if the angle of elevation at B is 22.3° then angle B in your triangle ABC is its supplement: 157.7°.
b = 1503 sin157.7° / sin5.4° = 6060
But in the triangle I was thinking of/the one I drew, the mountain is inside triangle ABC...? I don't really understand why I would have to take the supplementary angle of 22.3; I'm not disagreeing with you, I can see that you're right, I just don't understand how and I want to understand for future reference.
Hmmm. What is C? I used C for the top of the mountain. What other point is of interest?
Dnt knw
To find the distance to the top of the mountain from point B, you will need to use the properties of triangles and trigonometric ratios.
Firstly, let's construct triangle ABC where A and B are the points on level ground, and C is the top of the mountain. The angles of elevation from points A and B are given as 16.9° and 22.3° respectively.
To calculate the distance to the top of the mountain from point B, we can use the trigonometric ratio tangent.
Let's call the distance to the top of the mountain from point B as x. Now, we have two right triangles: triangle ABD and triangle BCD. We can use the tangent function to relate the side opposite the angle of elevation (CD) in triangle BCD to the side adjacent to the angle of elevation (BD):
tan(22.3°) = CD/BD
Simplifying this equation, we get:
CD = BD * tan(22.3°)
Now, we need to find BD. To do this, we can use the sine function in triangle ABD:
sin(16.9°) = BD/AB
Rearranging the equation, we get:
BD = AB * sin(16.9°)
Now, we can substitute this value of BD into the previous equation:
CD = (AB * sin(16.9°)) * tan(22.3°)
Given that AB is 1503 m, substitute this value into the equation:
CD = (1503 * sin(16.9°)) * tan(22.3°)
Using a calculator, evaluate this expression to find the value of CD, which represents the distance to the top of the mountain from point B.