john is standing 100 metres away from a building.the angle of elevationto the top of the building is8 degrees .He now walks straight away from the building.When he stopes,the angle of elevation is only 1 degree and 48 seconds.How far away is now from the base of the building.
The height H of the building can be found from the equation:
tan(8°)=H/100
or
H=100tan(8°) ....(1)
After he "backed-up" to a distance of D from the building, the new relation with the height is:
tan(1°48')=H/D
or
D=H/tan(1°48')....(2)
So substitute H from (1) into (2) to get
D=100tan(8°)/tan(1°48')
Can you take it from here?
(The answer is around 450m)
To solve this problem, we can use trigonometry, specifically the tangent function, to find the distances involved.
Let's consider the first situation where John is standing 100 meters away from the building and the angle of elevation to the top of the building is 8 degrees.
From the problem, we can infer that John is standing on level ground. Therefore, we have a right-angled triangle, with the distance from John to the base of the building as the adjacent side, the height of the building as the opposite side, and the angle of elevation as the angle.
Using the tangent function, we can calculate the height of the building as follows:
tangent(angle) = opposite / adjacent
tangent(8 degrees) = height / 100 meters
To find the height, we can rearrange the equation:
height = tangent(8 degrees) * 100 meters
Now, let's move on to the second situation where John, after walking straight away from the building, stops, and the angle of elevation is only 1 degree and 48 seconds.
Again, we have a right-angled triangle with the distance from John to the base of the building (let's call it "x") as the adjacent side, the same height of the building as before as the opposite side, and the new angle of elevation as the angle.
Using the same logic, we can set up the following equation:
tangent(angle) = opposite / adjacent
tangent(1 degree and 48 seconds) = height / x meters
However, it's easier to work with degrees, so let's convert the angle from degrees and seconds to decimal degrees.
1 degree = 1
48 seconds = 48/60 = 0.8 degrees (since there are 60 seconds in a minute)
Now, we can calculate the distance from John to the base of the building as follows:
x = height / tangent(angle)
x = height / tangent(1.8 degrees)
Now, substitute the value of height we calculated earlier:
x = (tangent(8 degrees) * 100 meters) / tangent(1.8 degrees)
Using a scientific calculator or trigonometric tables, calculate the tangent values for both angles and substitute them into the equation to find the value of x.