The top of a building 24cm high is observed from the top and from the bottom of a vertical tree. The angles of elevation are found to be 45degree and 60degree respectively . Find the height of the tree. ( leave your answer in surd form)
I need the workings
to know it
I don't understand the question
please show the diagram
assume you mean 24 m not cm
distance d from tree
tan 45 = (24-h)/d
tan 60 = h/d
so
(24-h)/tan 45 = h/tan 60
To find the height of the tree, we can use trigonometry. Let's call the height of the tree "h."
From the top of the tree, the angle of elevation to the top of the building is 45 degrees. This means that if we draw a right triangle with the tree height as the vertical side and the distance from the person to the tree as the horizontal side, the angle between these sides is 45 degrees.
Similarly, from the bottom of the tree, the angle of elevation to the top of the building is 60 degrees. This means that if we draw another right triangle with the tree height as the vertical side and the distance from the person to the tree as the horizontal side, the angle between these sides is 60 degrees.
Now, let's use the tangent function to relate the angle and the sides of a right triangle:
For the triangle at the top:
tan(45 degrees) = h / x, where x is the distance from the person to the tree.
For the triangle at the bottom:
tan(60 degrees) = h / (x + 24), where x + 24 is the distance from the person to the tree.
Using trigonometric identities, we know that tan(x) = sin(x) / cos(x). Let's rewrite the equations accordingly:
For the triangle at the top:
h / x = sin(45 degrees) / cos(45 degrees)
For the triangle at the bottom:
h / (x + 24) = sin(60 degrees) / cos(60 degrees)
Simplifying further:
h / x = (√2 / 2) / (√2 / 2) = 1
h / (x + 24) = (√3 / 2) / (1 / 2) = √3
Now, we have two equations:
h / x = 1 ---- (1)
h / (x + 24) = √3 ---- (2)
Let's solve these equations simultaneously:
From (1), we get h = x.
Substituting h = x in equation (2):
x / (x + 24) = √3
Cross multiplying, we get:
x = (√3)(x + 24)
Expanding the right side:
x = √3x + 24√3
Rearranging the equation to isolate x:
√3x - x = 24√3
(√3 - 1)x = 24√3
x = (24√3) / (√3 - 1)
Now, we can substitute this value of x into equation (1) to find h:
h = x = (24√3) / (√3 - 1)
Therefore, the height of the tree is (24√3) / (√3 - 1).