An aeoroplane flying at a ht 3000 mtr from the ground passes vertically above another plane at an instant angle of elevation from the ground are 60 and 30 degrees respectivly find distance between the planes
Assuming the angles are measured from the same point on the ground, the distance is
3000tan60° - 3000tan30°
tan 60o = (Y1+Y2)/X1 = 3000/X1
X1*tan 60 = 3000
X1 = 3000/tan60 = 1732 m.
tan 30o = (Y1+Y2)/(X1+X2)=3000/(1732+X2)
(1732+X2)*tan30 = 3000
1732+X2 = 3000/tan30 = 5196
X2 = 5196-1732 = 3464 m
tan30 = Y2/X2 = Y2/3464
Y2 = 3464*tan30 = 2,000 m.
Y1 = 3000-2000 = 1,000 m. = Distance between the planes.
To find the distance between the planes, we can use trigonometry and create a right triangle.
Let's label the variables:
- The height of the first plane from the ground: h1 = 3000 meters
- The angle of elevation of the first plane: θ1 = 60 degrees
- The angle of elevation of the second plane: θ2 = 30 degrees
- The distance between the planes: d (what we need to find)
Now let's look at the situation diagrammatically:
```
/|
h2/ |
/ |d
/ |
/θ2 |
/____|
/θ1 h1
```
Since both planes are flying vertically above each other, the line connecting them (labeled as 'd') would be parallel to the ground. Hence, we can consider the triangles ΔABC and ΔADE similar.
Using trigonometry, we can write down the following equations for both triangles:
In ΔABC:
tan(θ1) = h2 / d ---(1)
In ΔADE:
tan(θ2) = (h1 + h2) / d ---(2)
Now we can solve these two equations simultaneously to find the value of 'd'.
Dividing equation (1) by equation (2):
tan(θ1) / tan(θ2) = (h2 / d) / ((h1 + h2) / d)
Canceling out 'd' on both sides:
tan(θ1) / tan(θ2) = h2 / (h1 + h2)
Now substituting the given values:
tan(60) / tan(30) = 3000 / (3000 + h2)
√3 / (1/√3) = 3000 / (3000 + h2)
√3^2 / 1 = 3000 / (3000 + h2)
3 = 3000 / (3000 + h2)
Cross-multiplying and rearranging:
3(3000 + h2) = 3000
9000 + 3h2 = 3000
3h2 = 3000 - 9000
3h2 = -6000
h2 = -2000 meters
Since we cannot have a negative height for the second plane, we made a mistake somewhere in our calculations, or the given information is not feasible. Please double-check the values provided and ensure that they are correct.