The angle of elevation of the top of a tower is 27 degrees from point A due east of it. The angle of elevation of the top of the tower is 11 degrees from another point B due south of the tower. Find the height of the tower if the distance AB is 40m.
Solving after steve's step:
(PQ/tan27°)^2 + (PQ/tan11°)^2 = 40^2
Take PQ^2 out as it is common to both:
PQ^2(1/tan27^2 + 1/tan11^2)= 1600
PQ^2=1600/(1/tan27^2 + 1/tan11^2)
PQ=√1600/(1/tan27^2 + 1/tan11^2)
PQ=7.26( 3 significant figures)
If we label the top of the tower T and the bottom Q, then we have
tan 27° = PQ/QA
tan 11° = PQ/QB
QA^2 + QB^2 = 40
(PQ/tan27°)^2 + (PQ/tan11°)^2 = 40^2
Now just solve for PQ, the height of the tower.
To find the height of the tower, we can use the concept of trigonometry.
Let's label the height of the tower as "h."
From point A, the angle of elevation to the top of the tower is 27 degrees. This means that we can form a right-angled triangle with the tower, point A, and a point C directly below the top of the tower. The side opposite the angle of elevation is the height of the tower (h), and the adjacent side is the horizontal distance AC.
Similarly, from point B, the angle of elevation to the top of the tower is 11 degrees. We can form another right-angled triangle with the tower, point B, and the same point C below the top of the tower. The side opposite the angle of elevation is again the height of the tower (h), and the adjacent side is the horizontal distance BC.
We are given that the distance AB is 40m, which means that AC + BC = 40m.
Now, let's use trigonometry to find the values of AC and BC.
For triangle ABC:
tan(27) = h/AC
AC = h/tan(27)
For triangle BAC:
tan(11) = h/BC
BC = h/tan(11)
We can substitute these values into the equation AC + BC = 40:
h/tan(27) + h/tan(11) = 40
Let's simplify this equation:
h(1/tan(27) + 1/tan(11)) = 40
h = 40 / (1/tan(27) + 1/tan(11))
Using a scientific calculator or software, we can find that:
tan(27) = 0.5095 and tan(11) = 0.1944
Substituting these values into the equation:
h = 40 / (1/0.5095 + 1/0.1944)
h = 40 / (1.9591 + 5.1429)
h = 40 / 7.102
h ≈ 5.638 m
Therefore, the height of the tower is approximately 5.638 meters.
To find the height of the tower, we can use trigonometric ratios and create two right triangles.
Let's consider triangle ABC, where A is the point to the east, B is the point to the south, and C is the top of the tower. The information given states that AB = 40m.
In triangle ABC, we have two angles of elevation:
- The angle of elevation at point A (to the east) is 27 degrees.
- The angle of elevation at point B (to the south) is 11 degrees.
To find the height of the tower, we need to find the length of BC (the "height" of the tower).
We can start by finding the length of AC (the hypotenuse of triangle ABC).
To find AC, we can use the sine ratio:
sin(angle of elevation) = opposite/hypotenuse
In triangle ABC, the sin(27°) = BC/AC (since BC is opposite the angle of elevation at A).
Rearranging the equation, we get:
AC = BC / sin(27°) -- (1)
Similarly, in triangle ABC, the sin(11°) = AB/AC (since AB is opposite the angle of elevation at B).
Rearranging the equation, we get:
AC = AB / sin(11°) -- (2)
Since we have two expressions for AC, we can set them equal to each other:
BC / sin(27°) = AB / sin(11°)
Plugging in the known values, we get:
BC / sin(27°) = 40 / sin(11°)
Now, we can solve for BC by rearranging the equation:
BC = (40 / sin(11°)) * sin(27°)
Using a calculator, the value of BC is approximately 88.3m (rounded to one decimal place).
Hence, the height of the tower (BC) is approximately 88.3m.