from a horizontal distance of 1.05km a plot observe that the angles of depression of the top and base of a control tower are 36 and 41 respectively. calculate the shortest distance between the pilot and the base of the control tower and calculate the highest of the control tower.
from a horizontal distance of 1.05km a plot observe that the angles of depression of the top and base of a control tower are 36 and 41 respectively. calculate the shortest distance between the pilot and the base of the control tower and calculate the highest of the control tower.
From a horizontal distance x, if the tower has height h and its top is y meters below the plane's height, we have
y/x = tan36°
y = x tan36° = 1050 tan36° = 763 m
(h+y)/x = tan41°
h = 1050 tan41° - 763 = 150 m
The distance z is
z^2 = 1050^2 + 913^2
To solve this problem, we can use the concept of trigonometry, specifically the tangent function. The tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the adjacent side.
Let's define the given values:
- The horizontal distance from the plot to the control tower base is 1.05 km. Let's call this distance "x."
- The angle of depression from the top of the control tower is 36°.
- The angle of depression from the base of the control tower is 41°.
Now, let's solve for the shortest distance between the pilot and the base of the control tower.
Step 1: Find the height of the control tower using the angle of depression from the base.
We can use the tangent function to find the height:
tan(41°) = height of control tower / x
Multiplying both sides by x:
x * tan(41°) = height of control tower
Step 2: Find the shortest distance between the pilot and the base of the control tower.
This can be calculated by subtracting the height of the control tower from the height of the top of the control tower (which can be found using the angle of depression from the top).
Using the tangent function:
tan(36°) = height of control tower / x
Multiplying both sides by x:
x * tan(36°) = height of control tower + height of the top of the control tower
Since we already know the height of the control tower from Step 1, we can substitute that value:
x * tan(36°) = x * tan(41°) + height of the top of the control tower
To isolate the height of the top of the control tower, subtract x * tan(41°) from both sides:
x * tan(36°) - x * tan(41°) = height of the top of the control tower
Now, we know the height of the control tower, the height of the top of the control tower, and the distance between the pilot and the base. We can use these values to calculate the required distances.
To find the shortest distance between the pilot and the base of the control tower, we substitute the value of the height of the control tower into the equation:
shortest distance = x * tan(36°) - x * tan(41°)
To find the height of the control tower, we substitute the value of x into the equation:
height of control tower = x * tan(41°)
You can now calculate the values using a scientific calculator or a trigonometric table.