The angles of elevation of an office building as observed from the top and ground level of a 10 meters tall residential building are 68° and 72°
respectively. How tall is the office building? How far apart are the two buildings?
make a sketch
label the top of the taller building A and its bottom as B
label the top of the residential building as C, and draw a horizontal line to hit AB at P
So now you have two right-angled triangles
notice that BP = 10
in the bottom triangle,
tan 72 = 10/CP
CP = 10/tan72° ----> the distance between the two buildings
in the top triangle:
tan 68 = AP/CP
AP = CPtan68
height of building = 10 + AP
I will let you do the button-pushing
3.25 and 18.04
215
To find the height of the office building, we can use trigonometry. Let's call the height of the office building 'h' and the distance between the two buildings 'd'.
Using the angle of elevation of 68°, we can form a right-angled triangle with the height of the 10 meters residential building. The office building's height 'h' will be the sum of the height of the residential building (10 meters) and the opposite side of the triangle.
Using the trigonometric function tangent (tan), we have:
tan(68°) = (h + 10) / d
Rearranging the equation, we get:
(h + 10) = d * tan(68°) --------(1)
Similarly, using the angle of elevation of 72°, we can form a right-angled triangle with the height of the residential building. The office building's height 'h' will be the difference between the height of the triangle and the height of the residential building.
Using the trigonometric function tangent (tan), we have:
tan(72°) = (h - 10) / d
Rearranging the equation, we get:
(h - 10) = d * tan(72°) --------(2)
We now have a system of two equations (equations 1 and 2) with two unknowns (h and d). We can solve this system of equations simultaneously to find the values of 'h' and 'd'.
Solving equations (1) and (2) simultaneously, we get:
d * tan(68°) = d * tan(72°) + 20
Simplifying further:
d * (tan(68°) - tan(72°)) = 20
d = 20 / (tan(68°) - tan(72°))
Now that we have found the value of 'd', we can substitute it back into either equation (1) or (2) to find the value of 'h'.
Let's use equation (1):
(h + 10) = d * tan(68°)
h = (d * tan(68°)) - 10
By substituting the value of 'd' into equation (1), we can calculate the height 'h' of the office building.
Similarly, the distance between the two buildings, 'd', can be obtained by substituting the value of 'h' into equation (2):
(h - 10) = d * tan(72°)
d = (h - 10) / tan(72°)
By substituting the value of 'h' into equation (2), we can calculate the distance 'd' between the two buildings.