from a window 25meters above a street,the angle of elevation of the top of a wall on the opposie side is 15'.If the angle of depression of the base of the wall from the window is 35',find:
(i)the width of the street.
(ii)the hieght of the wall on the opposite side.
I sketched a diagram following your description to get
tan 35° = 25/(width of street)
width of street = 25/tan35 = 35.7 m
tan 15° = (top part of wall)/35.7
top part of wall = 35.7tan15 = 9.57
height of whole wall = 25 + 9.57 = 35.57 m
To find the width of the street and the height of the wall on the opposite side, we can use trigonometric ratios and basic geometry principles.
(i) The width of the street:
Let's label the width of the street as 'x'.
We have an angle of depression of 35' from the window. This angle is formed by a line of sight from the window to the base of the wall and a horizontal line passing through the window.
Using the tangent ratio, we can set up the following equation:
tan(35') = x / 25
Simplifying the equation, we have:
x = 25 * tan(35')
Using a scientific calculator, we find:
x ≈ 14.92 meters
Therefore, the width of the street is approximately 14.92 meters.
(ii) The height of the wall on the opposite side:
Let's label the height of the wall as 'h'.
We have an angle of elevation of 15' from the window. This angle is formed by a line of sight from the window to the top of the wall and a horizontal line passing through the window.
Using the tangent ratio again, we can set up the following equation:
tan(15') = h / x
Substituting the value of x from the previous step, we have:
tan(15') = h / 14.92
Rearranging the equation, we get:
h = 14.92 * tan(15')
Using a scientific calculator, we find:
h ≈ 4.29 meters
Therefore, the height of the wall on the opposite side is approximately 4.29 meters.
To solve this problem, we'll use the properties of angles of elevation and depression. Let's break it down step by step:
(i) Finding the width of the street:
First, let's draw a diagram to visualize the situation.
```
|
|
25m | Wall
|
|
-----------------|----------------
Street |
|
```
From the given information, the angle of depression from the window to the base of the wall is 35 degrees. This means that the angle formed between the horizontal line and the line of sight from the window to the base of the wall is 35 degrees.
Since we're given the height (25 meters) and the angle of depression (35 degrees), we can use trigonometry to find the width of the street (let's call it x).
Using the tangent function, we have:
tan(35) = height of the window / width of the street
Therefore, we can solve for x:
x = height of the window / tan(35)
x = 25 / tan(35)
Calculating this, we find:
x ≈ 25 / 0.7002
x ≈ 35.69 meters
So, the width of the street is approximately 35.69 meters.
(ii) Finding the height of the wall:
Again, let's draw a diagram to visualize the situation:
```
|
|
25m | Wall
|
|
-----------------|----------------
Street |
|
```
We're given the angle of elevation from the window to the top of the wall as 15 minutes (represented as 15').
Now, 1 degree can be divided into 60 minutes ('). So, 15 degrees is equal to 15 * 60 = 900 minutes.
Using trigonometry, we can find the height of the wall:
tan(angle of elevation) = height of the wall / distance to the wall
tan(15') = height of the wall / distance to the wall
Since we know the distance to the wall is the sum of the width of the street and the distance from the window to the wall, we can write:
tan(15') = height of the wall / (width of the street + distance from the window to the wall)
Substituting the known values:
tan(15') = height of the wall / (35.69 + 0)
tan(15') = height of the wall / 35.69
To find the height of the wall, multiply both sides by 35.69:
height of the wall = 35.69 * tan(15')
Calculating this, we find:
height of the wall ≈ 35.69 * 0.2618 (using a calculator)
height of the wall ≈ 9.35 meters (rounded to two decimal places)
Therefore, the height of the wall on the opposite side is approximately 9.35 meters.