from a window 30ft above the ground, the angle of elevation to the top of the building across the street is 50° and the angle of depression to the base of the building is 20° find the height of the building across the street
I am sure you made a sketch.
On mine, I labeled the window position W, the top of the building across the street T and the bottom of the building B. I also labeled the point where the horizontal from W meets TB as M
In triangle WBM,
tan20° = 30/WM
WM = 30/tan20°
in triangle WMT ,
tan50° = TM/WM
TM = WMtan50 = (30/tan20)(tan50)
= ...
TB = 30 + TM = ...
you do the button-pushing
To find the height of the building across the street, we can use trigonometry. Let's break down the problem and solve it step by step.
Step 1: Draw a diagram:
Start by drawing a diagram that represents the problem. Consider a window 30ft above the ground, the top of the building across the street, and the base of the building.
Step 2: Label the diagram:
Label the angle of elevation as 50° and the angle of depression as 20°. Also, label the height of the window as 30ft and the height of the building as h (the value we want to find).
Step 3: Identify the right triangle:
In the diagram, we have a right triangle formed by the height of the window, the distance across the street, and the height of the building.
Step 4: Identify the trigonometric ratios:
Since we are given two angles, we can use the trigonometric ratios of sine, cosine, and tangent.
Step 5: Use the trigonometric ratios:
We can use the angle of elevation and the height of the window to find the distance across the street (adjacent side of the triangle).
tan(50°) = opposite/adjacent
tan(50°) = 30/adjacent
Solving for the adjacent side:
adjacent = 30/tan(50°)
Similarly, we can use the angle of depression and the height of the window to find the distance from the window to the base of the building (opposite side of the triangle).
tan(20°) = opposite/adjacent
tan(20°) = 30/adjacent
Solving for the adjacent side:
adjacent = 30/tan(20°)
Step 6: Calculate the height of the building:
The height of the building is equal to the sum of the height of the window and the height from the base of the building to the ground.
height of building = height of window + (distance across the street - distance to the base of the building)
Substituting the values we found in steps 5:
height of building = 30 + (30/tan(50°) - 30/tan(20°))
Now you can use a calculator to find the height of the building across the street.