Please help me on this math problem

Question

Alex is 1.80 meters tall. He casts a shadow that is 1.45 meters long. At a distance of 4.95 meters from the tree his shadow reaches the end of a tree's shadow. What is the tree's height?

Image provided: https://ibb.co/pzmfyVj

Answer 1

We can think of this problem as two triangles. Because we know the entire length of the tree to Alex's shadow (4.95 + 1.45 = 6.4 m). Therefore, all we need is to find the angle made between Alex and his shadow so we can apply it to find the tree's lenght via the tangent function.

On the smaller triangle, we know Alex's height is 1.8 m and the distance between him and the shadow is 1.45 m. Therefore, we can set up a tangent function because one side is opposite to the common angle in both triangles (the bottom right) and the other is the adjacent side as it is next to the angle and is not directly across from the right angle. Setting up this equation:

tanθ = 1.8/1.45

Now we must take the inverse tangent function to find the angle

tan^-1(1.8/1.45) = θ

θ = 51.147°

Now that we have an angle and the adjacent side, we can apply the tangent function to find the height of the tree (x ≡ height of the tree):

tan(51.147°) = x/6.4m

x = tan(51.147°) • 6.4m

x = 7.94 m

The tree is 7.94 m tall