A 2 m tall person standing on the ground looks up at an angle of 34.0 o with the horizontal and sees the top of a tree and the top of a building aligned. The tree is 20 m away from the person, and the building is 51.0 m from the person
To find the height of the building, we can use trigonometry and the concept of similar triangles.
Let's label the height of the building as h, the height of the tree as t, the distance from the person to the tree as d1, and the distance from the person to the building as d2.
We know that the person's line of sight is making an angle of 34.0° with the horizontal. This means that the angle between the person's line of sight and the direction straight up to the top of the tree is also 34.0°. Similarly, the angle between the person's line of sight and the direction straight up to the top of the building is also 34.0°.
Now, we can set up a proportion using the similar triangles formed by the person, the tree, and the building:
t / d1 = (t + h) / d2
Since we know that the height of the person is 2 m, we can rewrite the equation:
t / 20 = (t + h) / 51
To solve for h, we can cross-multiply:
51t = 20(t + h)
51t = 20t + 20h
Subtracting 20t from both sides:
31t = 20h
Dividing both sides by 20:
h = (31/20) * t
So, to find the height of the building (h), we need to find the height of the tree (t) first.
To find the height of the tree, we can use the trigonometric function tangent (tan):
tan(34.0°) = t / 20
t = 20 * tan(34.0°)
Using a calculator:
t ≈ 11.976 m (rounded to three decimal places)
Finally, substituting the value of t into the equation for h:
h = (31/20) * 11.976
Using a calculator:
h ≈ 18.542 m (rounded to three decimal places)
Therefore, the height of the building is approximately 18.542 meters.