From an aeroplane in the air and at a horizontal distance of 1050m, the angles of depression of the top and base of a control tower at an instance are 〖36〗degrees and 〖41〗degrees respectively. Calculate, correct to the nearest metre, the
Height of the control tower
Shortest distance between the aeroplane and the base of the control tower.
If the height of the tower is h, and the plane is x m above the top of the tower,
x/1050 = tan36°
(h+x)/1050 = tan41°
Now you have h and x, and the distance in question is
d^2 = 1050^2 + (x+h)^2
To find the height of the control tower, we will use trigonometry.
Let's call the height of the control tower "h" and the shortest distance between the aeroplane and the base of the control tower "d".
Using the angle of depression, we can create a right triangle with the height of the control tower as the vertical side and the horizontal distance from the aeroplane to the control tower as the base. We can use the tangent function to relate the angle and the sides of the triangle.
The tangent function is defined as the opposite side divided by the adjacent side:
tan(angle) = opposite / adjacent
For the top angle (36 degrees):
tan(36) = h / 1050
Rearranging the equation, we get:
h = tan(36) * 1050
Using a calculator, we can calculate the value of tan(36) as approximately 0.7265.
So, the height of the control tower (h) is:
h = 0.7265 * 1050
h ≈ 763.3 meters
Therefore, the height of the control tower is approximately 763.3 meters.
To find the shortest distance between the aeroplane and the base of the control tower (d), we can create another right triangle using the angle of depression for the base of the control tower.
Now, the opposite side is the height of the control tower (h) and the adjacent side is the shortest distance between the aeroplane and the base of the control tower (d).
Using the tangent function, we have:
tan(angle) = opposite / adjacent
For the base angle (41 degrees):
tan(41) = h / d
Rearranging the equation, we get:
d = h / tan(41)
Using the calculated value of h from before and the value of tan(41) as approximately 0.8692 (using a calculator), we can calculate d:
d = 763.3 / 0.8692
d ≈ 879.1 meters
Therefore, the shortest distance between the aeroplane and the base of the control tower (d) is approximately 879.1 meters.
To summarize:
- The height of the control tower is approximately 763.3 meters.
- The shortest distance between the aeroplane and the base of the control tower is approximately 879.1 meters.
To solve this problem, we can use trigonometry and apply the tangent function to find the height of the control tower and the shortest distance between the airplane and the base of the control tower.
Let's break down the given information:
Angle of depression to the top of the control tower = 36 degrees
Angle of depression to the base of the control tower = 41 degrees
Horizontal distance between the airplane and the control tower = 1050m
First, let's calculate the height of the control tower:
Step 1: Calculate the distance from the airplane to the top of the control tower.
We can use the tangent of the angle of depression to find it:
Distance to the top = tan(36 degrees) * 1050m
Step 2: Calculate the distance from the airplane to the base of the control tower.
Using the same logic, we can use the tangent of the angle of depression to find it:
Distance to the base = tan(41 degrees) * 1050m
Step 3: Calculate the height of the control tower.
The height of the control tower is the difference between the distances to the top and the base:
Height = Distance to the top - Distance to the base
Now, let's calculate the shortest distance between the airplane and the base of the control tower:
Step 4: Calculate the vertical distance from the airplane to the base of the control tower.
The vertical distance is simply the height of the control tower, which we already calculated.
Step 5: Calculate the shortest distance using the Pythagorean theorem.
The shortest distance is the hypotenuse of a right triangle with the vertical distance as one side and the horizontal distance as the other side:
Shortest distance = sqrt((Distance to the base)^2 + (Horizontal distance)^2)
Let's calculate the height of the control tower and the shortest distance:
Distance to the top = tan(36 degrees) * 1050 = 621.699 m (approx.)
Distance to the base = tan(41 degrees) * 1050 = 737.884 m (approx.)
Height = 621.699 m - 737.884 m = -116.185 m (approx.) [Negative sign indicates that the control tower is lower than the airplane.]
Shortest distance = sqrt((737.884 m)^2 + (1050 m)^2) = 1378.417 m (approx.)
Therefore, the height of the control tower is approximately -116.185 m, and the shortest distance between the airplane and the base of the control tower is approximately 1378.417 m.