The top of a building 24cm high is observed from the top and from the bottom of a vertical tree .the angle of elevations are found to be 45 and 60 respectively find the height of the tree

A building 24 cm high???

Anyway, I will let you decide what units you are using.

Distance of tree from building ---- x
tan 60° = 24/x
x = 24/tan60° = ....

height of tree ---- h
tan 45° = (24-h)/x
1 = (24-h)/x
x = 24-h

solve for h

To find the height of the tree, we can use right triangle trigonometry. Let's denote the height of the tree as 'h'.

From the top of the building, the angle of elevation is 45 degrees. This means that the tangent of the angle is equal to the height of the tree (h) divided by the distance between the top of the building and the tree (d).

Using the tangent function:

tan(45) = h/d

Since the angle of elevation from the bottom is 60 degrees, we can use the same equation:

tan(60) = h/(d+24)

We can now solve these two equations to find the value of 'h'.

tan(45) = h/d
1 = h/d
h = d

tan(60) = h/(d+24)
√3 = h/(d+24)

Since we have h = d from the first equation, we can substitute this into the second equation:

√3 = d/(d+24)

Now, we can solve this equation for 'd'.

√3(d+24) = d
√3d + 24√3 = d
24√3 = d - √3d
24√3 = (1 - √3) * d
24√3/(1 - √3) = d

Simplifying this expression:

d = 24√3/(1 - √3)

Now that we have the value of 'd', we can find the height of the tree by substituting this value into either of the original equations:

h = d = 24√3/(1 - √3)

Therefore, the height of the tree is 24√3/(1 - √3) centimeters.

To find the height of the tree, we can use trigonometry and set up a right triangle.

Let's consider the top of the building as point A, the top of the tree as point B, and the bottom of the tree as point C. The height of the building is 24 cm.

We are given two angles of elevation, one from the top of the tree looking at the top of the building (angle B) and one from the bottom of the tree looking at the top of the building (angle A).

Angle A = 45° (angle of elevation from the top of the tree to the top of the building)
Angle B = 60° (angle of elevation from the bottom of the tree to the top of the building)

Let's label the height of the tree as 'x'.

Now, we can use the tangent function to solve for x:

tan(A) = opposite/adjacent
tan(45°) = 24/x

Rearranging the equation, we have:
x = 24/tan(45°)

Next, let's use the tangent function for angle B to solve for the height of the tree:

tan(B) = opposite/adjacent
tan(60°) = (x + 24)/x

Rearranging the equation, we have:
x + 24 = x * tan(60°)
x + 24 = x * √3

To isolate x, we can subtract x from both sides of the equation:
24 = x * √3 - x

Next, we can factor x out of the right side of the equation:
24 = x (√3 - 1)

Finally, divide both sides of the equation by (√3 - 1) to get the value of x:
x = 24 / (√3 - 1)

Using a calculator, the value of x is approximately 41.57 cm.

Therefore, the height of the tree is approximately 41.57 cm.

This must be a typo -- a building 24 cm high.

24 centimeters is about the length of my arm from my elbow to my hand.