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January 20, 2009

Shadow length



A pillar 6 feet tall casts a shadow 3 feet long on the ground.
If the pillar was 12 feet tall, how many feet in length
would the shadow be ?

7 comments:

Secret Squïrrel said...

Assuming that the pillar is a cylinder, simple trigonometry gives 6ft as the answer.

Evan said...

This is a great question:

And the answer is... Unknown!

The shadow in this question is created as a result of the location of the light source. The light source is not specified to be the sun. If the location of the light source does not change position when we increase the height of the pilar, then the shadow will not be of equivalent length. Therefore, the shadow must be greater than 6ft.

To further elaborate, if the height of the 12 ft. pilar was equal to the height of the light source then the shadow becomes infinite.

Thus, using trigonometry, we can then calculate (tan^-1 x [6/3]) the angle to the light source (63.43494 degrees) but we do not know the height or the horizontal distance to it. Without either piece of information, The correct answer cannot be known, but it must be greater than 6.

(Note: even if we assume that the light source is the sun, then the shadow length becomes very, very, very close (within a fraction of a millimeter close) to 6 ft. but will not be exactly 6. The shadow must be greater than 6 ft.)

Anonymous said...

What a brilliant response! He is exactly correct. You must know where the light is comming from to answer the question.

Secret Squïrrel said...

Yeah, sure. If you want to be pedantic about it, you also need to know the shape of the surface upon which the shadow will be projected. Perhaps the ground slopes slightly upwards or maybe the pillar is on the Moon which has a smaller radius than the Earth.

But really, you're just being silly, aren't you Evan? If puzzle-setters had to specify absolutely every single parameter to satisfy nit-picking trolls like you, each statement of even a simple problem would run to a small essay - ridiculously unwieldy and a waste of time.

Ordinary puzzle-solvers understand that the shadow is being cast by the Sun and that the measurements take place on Earth such that the light rays can be considered to be parallel and the surface flat.

You need to find something constructive to do.

Anonymous said...

You'd assume the answer would be 6ft.
But Evan does have a point.

Anonymous said...

agree with secret squirrel, because the sun is so far away, one can treat the rays as parralell and because the curvature of the earth is so slight it can be treated as flat; evan would have a point if the sourse were significantly closer

SuShAnT said...

the shadow height is 3/6 i.e., 1/2 of pillar. Hence if the pillar is 12ft then the shadow would be 6ft.