Find the height of a tree when it is found that on walking away from it 20 metres, in horizontal line through its base, the elevation of its top changes from 60 degree to 30 degree.
To find the height of the tree, we can use trigonometry and the concept of similar triangles.
Let's assume the height of the tree is 'h' meters.
From the given information, when we walk away from the tree by 20 meters, we create a right-angled triangle with the base being the horizontal distance and the height being the difference in elevation from the top of the tree.
Using trigonometry, we can find the height of this right-angled triangle. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
We know that the tangent of 60 degrees is equal to the height of the tree divided by the distance from the tree. Similarly, the tangent of 30 degrees is equal to the height of the triangle we created divided by 20 meters (the horizontal distance).
So, we can write the equation as:
tan(60) = h / distance from the tree
tan(30) = height of triangle / 20
Using the value of tan(60) ≈ 1.732 and tan(30) ≈ 0.577, we can solve these equations to find the height of the tree.
1.732 = h / distance from the tree
0.577 = height of triangle / 20
Rearranging the first equation, we have:
h = 1.732 * distance from the tree
Substituting the value of h in the second equation, we get:
0.577 = (1.732 * distance from the tree) / 20
Now, we can solve this equation to find the distance from the tree.
0.577 * 20 = 1.732 * distance from the tree
11.54 = 1.732 * distance from the tree
Dividing both sides by 1.732, we get:
distance from the tree = 6.67 meters
Thus, the distance from the tree is approximately 6.67 meters.
To find the height of the tree, we substitute this distance back into the first equation:
h = 1.732 * distance from the tree
h = 1.732 * 6.67
h ≈ 11.53 meters
Therefore, the height of the tree is approximately 11.53 meters.
Let the height be H.
Let the (closer) distance for which the elevation is 60 degrees be D.
H/D = tan 60 = sqrt3 = 1.732
H/(D+20) = tan30 = 1/sqrt3= 0.5774
(D+20)/D = 3
1 + 20/D = 3
20/D = 2
D = 10 m, exactly
H = sqrt3*10 = 17.32 m