From a window A,10m above the ground, the angle of elevation of the top C of a tower is x degrees,where tan X=5/2 and the angle of depression of the foot D of the tower is y degrees, where tan y=1/4.calculate the height Cd of the tower in metres.
To calculate the height Cd of the tower, we can use the trigonometric properties of tangents and the given angles.
Let's break down the given information:
Angle of elevation from window A to top C of the tower:
tan X = 5/2
Angle of depression from window A to foot D of the tower:
tan y = 1/4
To solve this problem, we'll use the properties of right triangles and the tangent function.
Step 1: Calculate the height of AD.
The height AD can be found by using the tangent function and the angle of depression y:
tan y = AD / 10 (since AD is the opposite side and 10m is the adjacent side)
1/4 = AD / 10
To solve for AD, we can cross-multiply:
AD = (1/4) * 10
AD = 2.5m
Step 2: Calculate the value of angle B.
Angle B is the angle between the AD and CD, and it can be found by subtracting the angle of depression (y) from 90 degrees.
B = 90 - y
B = 90 - (1/4) * 180
B = 90 - 45
B = 45 degrees
Step 3: Calculate the length of CD using tangent function.
Using the tangent function and the angle B, we can find the length of CD:
tan B = CD / AD
Substitute the values:
tan 45 = CD / 2.5
Since tan 45 is equal to 1:
1 = CD / 2.5
To solve for CD, we can cross-multiply:
CD = 2.5m
Therefore, the height Cd of the tower is 2.5 meters.
Tan x = 5/2 = 2.50, X = 68.2o.
Tan y = 1/4 = 0.25, Y = 14.0o.
Tan 14 = 10/d1, d1 = 40.1 m. = Distance from bottom of window to foot of tower.
Tan 68.2 = d2/40.1, d2 = 100.3 m.
= Vert. distance from top of window to top of tower.
h = 10m + d2 = 10 + 100.3 = 110.3 m.