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The Truman Show (Posted on 2018-03-02) Difficulty: 3 of 5
Truman suspects that bright light in the sky may not actually be the sun. What if instead of millions of kilometers away, it is merely hundreds of meters up and attached to an invisible ceiling 'sky'?

He devises a shadow measuring experiment.

He stands a 1 meter stick and a 2 meter stick up vertically next to each other.

The 1 meter stick casts a shadow 3.46 meters long but the 2 meter stick casts a shadow 6.94 meters long. That's more than twice as long!

How high and far is the 'sun'?

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Solution solution with error analysis | Comment 2 of 4 |
Putting the "sun" at the right side and letting the tip of the longer shadow be the origin:

The difference between the two shadow lengths is 3.48 m.

y = (2/6.94)x
y = (1/3.46)(x-3.48)

(2/6.94 - 1/3.46)x = -3.48/3.46

x = (3.48/3.46) / (1/3.46 - 2/6.94) = 1208

y = (2/6.94)(1208) = 348

The "sun" is 348 meters (about 116 stories) up and 1208 meters away.

Even with the rounding to integers, we probably have too much precision. Suppose the shorter shadow were really 3.456 and the longer one 6.944, still rounding to the given figures. But computed exactly we'd get

y = (2/6.944)x
y = (1/3.456)(x-3.488)

(2/6.944 - 1/3.456)x = -3.488/3.456

x = (3.488/3.456) / (1/3.456 - 2/6.944) = 757

y = (2/6.944)(757) = 218

only 73 stories up, 757 meters away.

Errors in the other direction would make it higher and further.

Later addition:

Calculating the other direction of error gives 868 meters high (289 stories) at 3010 meters distance (about 3 km).

Edited on March 2, 2018, 11:00 am
  Posted by Charlie on 2018-03-02 10:32:54

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