A man is walking away from a lamppost with a light source h = 6 m above the ground. The man is m = 2 m tall.

Question

A man is walking away from a lamppost with a light source h = 6 m above the ground. The man is m = 2 m tall. How long is the man's shadow when he is d = 14 m from the lamppost? [Hint: Use similar triangles.] 

Do we use the d*h or is it d/h? I'm really confused as to how this question should be approached. Thanks

Answer 1

Don't think about this in terms of formulas! Think logically :)

Use this image for reference (https://www.nextgurukul.in/media/images/q2aanswers/5393_files/image001.jpg), but disregard the lengths.

1. Line segment AB represents the height of the lamp post.
2. Line segment DE represents the height of the man.
3. Line segment BD represents the distance d.
4. Line segment DC represents the man's shadow.

Constructing this is essential to the problem. The light from lamppost casts the furthest shadow from shining on the man's head (E) and hitting the ground (C).

Solution 1: Similar triangles

From here, we have two similar triangles: triangle CDE and triangle ABC.

They have similar adjacent and opposite sides. Compare CD with BC and ED with AB.

With this similarity, we can make a proportion, since the same angles θ is shared:

CD/ED = BC/AB

(small triangle vs. bigger triangle)

x/2 = (x+14)/6

Solving for x:

6x/2 = x + 14

3x = x + 14

2x = 14

x = 7

Therefore, the man's shadow is 7 m long.

Solution 2: Trigonometry with tangent

tan θ = CD/ED = BC/AB

x/2 = (x+14)/6

Solve for x: x = 7.

Answer 2

X is length of man's shadow

3x = x+14

2x=14

X=7