the elevation of a tower from a point on the ground is 30 at a point on the horizontal line passing through the foot of the tower and 100 metres nearer it, the angle of elevation is 60 find height of the tower and the distance of the first point from the tower
As always, draw a diagram. If the height is h, and the distance is d, you have
h/d = tan 60°
h/(d+100) = tan 30°
Now just solve for d and h.
To find the height of the tower and the distance of the first point from the tower, we can use trigonometry. Let's denote the height of the tower as 'h' and the distance of the first point from the tower as 'd'.
First, let's draw the situation described.
A
/|
h / | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
B /____________|_____________\
d C
From the given information, we can identify two right-angled triangles: Triangle ABC and Triangle BAC.
For Triangle ABC:
- The elevation of the tower from point A is 30 degrees.
- The distance from point A to point C is 'd' meters.
- The elevation of the tower from point B is 60 degrees.
For Triangle BAC:
- The elevation of the tower from point B is 60 degrees.
- The distance from point B to point C is 'd - 100' meters.
Using trigonometry, we can set up the following equations:
For Triangle ABC:
tan(30 degrees) = h / d
For Triangle BAC:
tan(60 degrees) = h / (d - 100)
Using these equations, we can solve for both 'h' and 'd'.
1. Solve for 'h':
Rearranging the equation for Triangle ABC, we get:
h = d * tan(30 degrees)
2. Solve for 'd':
Rearranging the equation for Triangle BAC, we get:
h = (d - 100) * tan(60 degrees)
Now, we can substitute the value of 'h' from equation 1 into equation 2:
d * tan(30 degrees) = (d - 100) * tan(60 degrees)
Simplify and solve for 'd':
d * √(3)/3 = (d - 100) * √(3)
√(3)d/3 = (√(3)d - 100√(3))
√(3)d/3 - √(3)d = -100√(3)
-d√(3)/3 = -100√(3)
d√(3)/3 = 100√(3)
d = 300 meters
Now substitute this value of 'd' into equation 1 to find 'h':
h = 300 * tan(30 degrees)
h = 300 * (√(3)/3)
h = 100√(3) meters
Therefore, the height of the tower is 100√(3) meters and the distance of the first point from the tower is 300 meters.