two observer p and r 15 meters apart observes a plane in the same vertical plane and from the same side of the kite,the angles of elevatio of the plane from p and r are 30 60 respectively.find the height of the plane to the nearest meter.
Let the plane's position be labelled A, and the bottom of the vertical B
In triangle APR, angle A = 30° using basic geometry.
Then PR = AR = 15, since we now have an isosceles triangle.
then sin60° = AB/AR
AB = 15sin60 = appr 12.99 or 13 metres
if the height of the plane is h, then
h cot30° - h cot60° = 15
Hor. distance of P = X meters.
Hor distance of R = x-15 meters.
Tan30 = h/x
h = x*Tan30.
Tan60 = h/(x-15)
h = (x-15)*Tan60.
h = x*Tan30 = (x-15)Tan60
0.58x = 1.73x-26
X = 23 m.
h = x*Tan30 = 23*0.58 = 13 m.
To find the height of the plane, we can use trigonometry and the concept of similar triangles.
Let's break down the problem step by step:
1. Draw a diagram: Draw a diagram representing the situation described in the problem. Label the observer points P and R, the distance between them as 15 meters, and the angles of elevation as 30° and 60° respectively.
2. Identify the known values: From the problem, we know that the distance between the observers P and R is 15 meters, and the angles of elevation from P and R are 30° and 60° respectively.
3. Establish a reference point: We will assume that the height of the plane is measured from the ground, so we can take a reference point below the observers (say, point O).
4. Determine the distance between O and the plane: Let's calculate the distance between O and the plane by using the tangent function. Since we have an angle of 30° from observer P, we can calculate the length of OP using the equation:
OP = tan(30°) * distance between observers (15 meters)
OP = tan(30°) * 15 meters
OP ≈ 8.66 meters (rounded to two decimal places)
5. Determine the distance between O and the plane using observer R: Similarly, we can calculate the length of OR using the angle of elevation from observer R. Since we have an angle of 60° from observer R, we can calculate the length of OR using the equation:
OR = tan(60°) * distance between observers (15 meters)
OR = tan(60°) * 15 meters
OR ≈ 25.98 meters (rounded to two decimal places)
6. Determine the height of the plane: Now, we can find the height of the plane by subtracting the length of OP from the length of OR:
Height of the plane = OR - OP
Height of the plane ≈ 25.98 meters - 8.66 meters
Height of the plane ≈ 17.32 meters (rounded to two decimal places)
Therefore, the height of the plane, to the nearest meter, is approximately 17 meters.