Explanation:
as always, such a problem has 2 solutions, because the teachers never give any information about the relative positions of the sun, the pole and the tree. is the sun shining first on the pole and then the tree, or is the sun shining first on the tree and then the pole ?
the difference is that in one case the shadow of the tree is 60 - 20 = 40 m, and in the other case it is 60 + 20 = 80 m long.
anyway, for a certain angle of the sunlight we know the shadow of the pole is 60 m long, which creates with the 4.8 m height a right-angled triangle, with the line of sight from the top of the pole to the ground end of the shadow being the Hypotenuse or radius for or trigonometric triangle in a circle.
the shadow length (60 m) is sine of the sunshine angle multiplied by the radius.
the height (4.8 m) is cosine of that sunshine angle multiplied also by that radius.
Pythagoras gets us the radius for the pole triangle :
radius² = 60² + 4.8² = 3600 + 23.04 = 3623.04
radius = sqrt(3623.04) = 60.19169378... m
sin(angle) × radius = 60
sin(angle) = 60/radius = 0.996815279...
angle = 85.42607874...°
the sunshine creates with the tree also a right-angled triangle with the same sunshine angle (the sun is so far away, that the tiny, tiny difference is irrelevant, we can simply assume the angles are equal).
but the shadow is of different length (sin(angle)×radius), which means also the radius for the tree triangle has to be different. and that defines the height of the tree (cos(angle)×radius).
but now we know the angle, and we can reverse calculate the side lengths.
but the question is (as mentioned at the beginning) : is the tree shadow 40 m or 80 m long.
we have therefore 2 solutions for the height of the tree.
1. the shadow is 40 m long.
sin(angle) × radius = 40
radius = 40/sin(angle) = 40.12779585... m
tree height = cos(angle)×radius = 3.2 m
2. the shadow is 80 m long.
sin(angle) × radius = 80
radius = 80/sin(angle) = 80.25559171... m
tree height = cos(angle)×radius = 6.4 m