From a point P on the ground the angles of elevation of the top of the building and of a helicopter, hovering at some height over the top of the building, are 30 degree and 60 degree, respectively. Find the height of the helicopter above the ground
draw a diagram.
If the building has height h, and
the chopper is x above the building
h cot60° = (h+x)cot30°
Both of these are equal to the distance of P from the building. Once you have that, you can find h and x.
Without knowing that or h, you cannot find x.
To find the height of the helicopter above the ground, we can use trigonometric ratios.
Let's denote the height of the building as 'h', the height of the helicopter above the top of the building as 'x', and the distance between the point P on the ground and the building as 'd'.
From the given information, we can form two right triangles: one with the building, the point P, and the top of the building, and another with the top of the building, the helicopter, and a point directly under the helicopter on the ground.
In the first triangle, the angle of elevation of the top of the building is 30 degrees. Therefore, the tangent of this angle is equal to the opposite side (h) divided by the adjacent side (d):
tan(30°) = h / d ---(Equation 1)
In the second triangle, the angle of elevation of the helicopter is 60 degrees. Therefore, the tangent of this angle is equal to the opposite side (h + x) divided by the adjacent side (d):
tan(60°) = (h + x) / d ---(Equation 2)
We have two equations with two unknowns (h and d), so we can solve them simultaneously.
First, solve Equation 1 for h:
h = d * tan(30°)
Next, substitute the value of h in Equation 2:
tan(60°) = (d * tan(30°) + x) / d
Simplify the equation:
√3 = tan(30°) + x / d
Now, multiply both sides by d:
√3 * d = tan(30°) + x
Subtract tan(30°) from both sides:
√3 * d - tan(30°) = x
Finally, substitute the given values to find the height of the helicopter above the ground:
x = √3 * d - tan(30°)
The height of the helicopter above the ground is √3 * d - tan(30°).