From a window in a building 20 m above ground the angle of depression of the top of a statue across the street is 30 degrees, and the angle of depression of the base of the statue is 34. How tall is the statue?
I hope you made a diagram.
label the window position P
label the top of the statue T
label the bottom of the statue B
So we want TB
I can find BP
sin34° = 20/BP
BP = 20/sin34
let's use the sine law in triangle TBP
TB/sin4 = BP/sin120
TB = BPsin4/sin120
= (20/sin34)(sin4/sin120)
= appr 2.88 m
To find the height of the statue, we can use the trigonometric concept of the angle of depression. Let's break down the problem step by step:
Step 1: Understand the problem.
We have a window that is 20 meters above the ground, and we observe the angles of depression of the top and base of the statue. We need to find the height of the statue.
Step 2: Visualize the problem.
To visualize the problem, draw a diagram. Represent the building, window, and the statue across the street as shown:
|\
| \
| \
| \ --- Statue
| \
|θ1 \
|---------------\
|----d-----|
The symbol θ1 represents the angle of depression of the top of the statue, and d represents the distance between the building and the statue.
Step 3: Identify the relevant trigonometric ratios.
Since we have the angles of depression, we can use the tangent ratio. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side of a right triangle.
In this case, the tangent of θ1 is equal to the height of the statue divided by the distance between the building and the statue (d). Let's write this equation:
tan(θ1) = height of statue / d
Step 4: Solve the equation.
From the problem, we know that θ1 is 30 degrees and θ2 (the angle of depression of the base of the statue) is 34 degrees. We also know that the height of the building is 20 meters.
Let's find the distance, d, between the building and the statue using the angle θ2:
tan(θ2) = height of building / d
tan(34) = 20 / d
To find d, rearrange the equation:
d = 20 / tan(34)
Now substitute the value of d into the equation we derived from step 3:
tan(θ1) = height of statue / (20 / tan(34))
Rearrange the equation and solve for the height of the statue:
height of statue = tan(θ1) * (20 / tan(34))
Step 5: Calculate the height of the statue.
Using a calculator, find the value of the tangent of angles θ1 and θ2, and substitute these values into the equation we derived in step 4:
height of statue = tan(30) * (20 / tan(34))
Calculating this expression will give you the height of the statue.