Yesterday afternoon the sun was shining into my room and making some shadows of different objects on my table. By looking at the shadows I decided to make a puzzle for this community. The picture below is what I arranged on my table and I want you to find the height of the pencil.
The height of the pencil is
To get this, I found what the length of the shadow would be if the wall were not there:
From the pencil sharpener, we know that the ratio between object height and shadow length is
2:3, that is, a 2cm object would create a 3cm shadow.
Using given lengths, the total height of the shadow is
h' = 8cm (3cm box and 5cm above the box)
That means that if the wall weren't there, the shadow would go
L' = 12cm past the wall (Note the similar triangles in my sketch above)
Using the lengths given, and the fact that the matchbox must be the same depth as the pencil sharpener (they produce the same length shadow), we can find what the total length of the shadow would be without the wall
L = 12cm + 8cm + 1cm = 21cm
Now, we just use the original ratio between shadow length and object height to find a final height of
h = 14cm
this is because the shadow has two parts, horizontal would be proportional, but vertical would be similar. By that I mean, the vertical portion of the shadow is parallel to the pencil, hence it must be generated from a length of pencil equal to itself.
vertical portion = 5cm + 3cm = 8cm (A)
Now for the horizontal portion, notice the width of the sharpner. The shadow is in the ratio of 1.5:1 w.r.t. the body.
Hence the horizontal portion of the pencil's shadow must also be in the same ratio. Since the horizontal portion of shadow is 8cm + 1cm = 9cm.
pencil portion casting horizontal shadow is 9/1.5 = 6cm (B)
Hence total length of pencil is:
(A) + (B) = 8 + 6 = 14 cm