A surveyor measures the angle of elevation of the top of the mountain from a point (P) at sea level as 30 detre along a track which slopes uniformly uphill towards the mountain. From this point, which is 50 metre above sea level, he measures the angle of elevation as 40 degree. (a) Find the value of x. (b) If PA = d m and AT = h m, express d in terms of h. (c) If AB = y m, express y in terms of h. (d) Find the height of the mountain above sea level, correct to the nearest meter.
To solve this problem, let's break it down step by step:
(a) Find the value of x:
We can use trigonometry to find the value of x. From the diagram, we have a right triangle PAB, where AB represents the distance from point A to the top of the mountain, and x represents the distance from point B to the top of the mountain.
Using the tangent function, we can write:
tan(40 degrees) = AB / 50
tan(40 degrees) = x / (50 + AB)
Substituting the value of AB from the first equation into the second equation, we get:
tan(40 degrees) = x / (50 + (tan(40 degrees) * 50))
Now, we can solve for x:
x = tan(40 degrees) * (50 + (tan(40 degrees) * 50))
Calculating this expression will give us the value of x.
(b) If PA = d m and AT = h m, express d in terms of h:
From the triangle PAT, we can see that d = h * tan(30 degrees). This is because the tangent function gives us the ratio of opposite (d) to adjacent (h). Therefore, d = h * tan(30 degrees).
(c) If AB = y m, express y in terms of h:
From the triangle PAB, we can see that y = h * tan(40 degrees). Similar to part (b), the tangent function gives us the ratio of opposite (y) to adjacent (h). Therefore, y = h * tan(40 degrees).
(d) Find the height of the mountain above sea level, correct to the nearest meter:
The height of the mountain above sea level is equal to the sum of the height at point A (50 meters) and the distance PB (x). Therefore, the height of the mountain above sea level is approximately 50 + x meters.