An aeroplane when 6000 m high passes vertically above another plane at an instant when their angles of elevation at the same observing point are 60° and 45° respectively. How many meters higher is the one than the other?
Draw a vertical line with A the top plane, B the lower plane and C the point on the ground below the planes.
ABC is a straight line.
Let D be the position of the observer on the ground
Let AB = x m
We know:
BC = 6000
∠BDC = 45°, so DC = 6000
∠DAC = 60°
in ∆ADC, tan 60° = (x+6000)/6000
(x+6000)/6000 = √3
x+6000 = 6000√3
x = 6000√3 - 6000 = 4392.3 m
It was useless
To determine how many meters higher one plane is than the other, we can use trigonometry and create a right triangle. Let's denote the height of the first plane as h1 and the height of the second plane as h2.
From the given information, we can determine the following:
- The angle of elevation to the first plane is 60°, so we have a right triangle with an angle of 60° and opposite side length h1.
- The angle of elevation to the second plane is 45°, so we have a right triangle with an angle of 45° and an opposite side length h2.
- The distance between the two observing points can be determined by subtracting the height of the second plane from the height of the first plane: h1 - h2 = 6000m.
Next, we can use the tangent function to relate the angles of elevation to the opposite side lengths in each triangle.
For the first plane:
tan(60°) = h1 / distance between the observing points
For the second plane:
tan(45°) = h2 / distance between the observing points
Since both triangles share the same observing point, they have the same distance between the observing points.
Rearranging the equations, we can solve for h1 and h2:
h1 = tan(60°) * distance between the observing points
h2 = tan(45°) * distance between the observing points
Since h1 - h2 = 6000m, we can substitute the expressions for h1 and h2:
tan(60°) * distance between the observing points - tan(45°) * distance between the observing points = 6000m.
Since the distance between the observing points is the same on both sides of the equation, we can factor it out:
(distance between the observing points) * (tan(60°) - tan(45°)) = 6000m.
Now, we can evaluate the difference between the tangent values:
tan(60°) - tan(45°) ≈ 1.732 - 1 = 0.732.
Plugging this back into the equation, we have:
(distance between the observing points) * 0.732 = 6000m.
(distance between the observing points) ≈ 6000m / 0.732 ≈ 8190.08m.
Therefore, the distance between the observing points is approximately 8190.08m. Since h1 - h2 = 6000m, we can subtract the two heights to find the difference:
h1 - h2 ≈ 8190.08m - 6000m ≈ 2190.08m.
Hence, the first plane is approximately 2190.08 meters higher than the second plane.
To find the difference in height between the two planes, you can use trigonometry and the concept of similar triangles.
Let's label the height of the first plane (the one with the angle of elevation of 60°) as h1, and the height of the second plane (the one with the angle of elevation of 45°) as h2.
We can set up two right-angled triangles, one for each plane, with the observing point as the vertex of the angle of elevation. The heights of the planes can be represented by the opposite sides of the triangles.
In the triangle for the first plane, we have an angle of 60° and a side opposite to that angle, which is the height h1. The adjacent side to the angle is the distance from the observing point to the first plane, which is 6000 meters.
In the triangle for the second plane, we have an angle of 45° and a side opposite to that angle, which is the height h2. The adjacent side to the angle is also the distance from the observing point to the second plane, which is 6000 meters.
Since the triangles share the same observing point, we can conclude that the two triangles are similar. This means that the ratios of corresponding sides are equal.
Using the tangent ratio, we can write the equations:
tan(60°) = h1 / 6000 --> (1)
tan(45°) = h2 / 6000 --> (2)
Now, let's solve these equations to find h1 and h2:
From equation (1), we can rearrange to find h1:
h1 = tan(60°) * 6000
Using a calculator, we can evaluate the tangents of 60°:
h1 = 1.732 * 6000
h1 ≈ 10392 meters
From equation (2), we can rearrange to find h2:
h2 = tan(45°) * 6000
Using a calculator, we can evaluate the tangents of 45°:
h2 = 1 * 6000
h2 = 6000 meters
Now, to find the difference in height between the two planes, we subtract h2 from h1:
Difference in height = h1 - h2
Difference in height = 10392 - 6000
Difference in height ≈ 4392 meters
Therefore, the first plane is approximately 4392 meters higher than the second plane.