From A, due east of the building, the angle of elevation of the top is 45 degree. From B, southwest of the building, the angle of elevation is 30 degree. If AB=250m, find the height of the building.
let C be the third vertex of the horizontal triangle, directly below the building
for the vertical triangles, height = AC(tan(45)) = BC(tan(30)) = AC, since tan(45) = 1
now use law of cosines on the horizontal triangle to obtain
250^2 = AC^2 + (AC/tan(30))^2 – 2(AC)(AC/tan(30))cos(135)
and solve the quadratic equation to obtain AC = height = 98.4413
jolly rancher, you are correct sir. Thank you sir
To find the height of the building, we can use trigonometry.
First, let's draw a diagram to represent the situation.
```
A
|\
h | \
| \
----\----- Building
x B
-------
250m
```
Let's label the height of the building as "h" and the distance from point A to the building as "x".
From point A, due east of the building, the angle of elevation of the top of the building is 45 degrees. This means that in a right-angled triangle formed by A, the top of the building, and the base of the building, the angle opposite to the side of length "x" is 45 degrees.
Similarly, from point B, southwest of the building, the angle of elevation of the top of the building is 30 degrees. This means that in a right-angled triangle formed by B, the top of the building, and the base of the building, the angle opposite to the side of length "250m - x" is 30 degrees.
Now, we can use trigonometry to set up two equations based on these right-angled triangles.
In triangle A, we have:
tan(45) = h / x
In triangle B, we have:
tan(30) = h / (250m - x)
We can solve these two equations simultaneously to find the value of "h".
Using the values of the tangent function for 45 degrees and 30 degrees, which are sqrt(2)/2 and sqrt(3)/3 respectively, we have:
sqrt(2)/2 = h / x
sqrt(3)/3 = h / (250m - x)
Now, we can solve these equations:
sqrt(2) * (250m - x) = sqrt(3) * x
Simplifying:
250sqrt(2)m - sqrt(2)x = sqrt(3)x
(250sqrt(2)m) = (sqrt(2) + sqrt(3))x
Now, we can solve for x:
x = (250sqrt(2)m) / (sqrt(2) + sqrt(3))
After calculating x, we can substitute this value back into either equation to find the height "h". I will use the equation from triangle A:
sqrt(2)/2 = h / x
After substituting the value of x, we can solve for "h":
h = (sqrt(2)/2) * [(250sqrt(2)m) / (sqrt(2) + sqrt(3))]
Finally, calculate the value of "h" to find the height of the building.