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What the problem says:

When a screen is placed $3 m$ from a projector, the picture occupies $3 m^2$. How large will the picture be when the projector is $5 m$ from the screen?

My direct answer would be $5 m^2$, but it seems too easy to be true. I am curious about what the answer would be, or how to do it - it is not something critical since I am doing this in my free time.

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Note that if you put it twice as far away, 6m, the image is not twice as big, $6m^2$, but four times, $12m^2$, because it is twice as wide and twice as tall. – MJD Mar 19 '12 at 19:54

The linear dimensions get multiplied by the factor $\frac{5}{3}$. So the area gets multiplied by $\left(\frac{5}{3}\right)^2$.

Added: There is a similar phenomenon when you enlarge a photograph. Suppose that you double the linear dimensions, changing a $4\times 5$ picture to a $8\times 10$ picture. Then the new area is $4$ times the original area. Multiplying the linear dimensions by $2$ multiplies area by $2^2$ (and multiplies volume by $2^3$).

If you have text or an image on a computer screen, and you scale by the factor $1.2$, the effect is quite dramatic, because areas are multiplied by $(1.2)^2$, so almost by $50$%.

There are all sorts of biological implications. For example, suppose that you take a regular person, and scale all of her/his linear dimensions by the factor $3$. So a $2$ metres tall person becomes $6$ metres tall. Note that her/his weight is $27$ times the previous weight. The strength of a bone is proportional to its cross-sectional area, so has only gone up by a factor of $9$. Thus a fall that would not seriously hurt the $2$ metre person could break the leg bones of the scaled up person. Heat loss is proportional to surface area, and heat production is proportional to mass. So the scaled up person would probably handle cold weather better. The same heat loss issue means that little mammals in a cool climate need to spend much of their time eating.

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I really appreciate your answer. However, I do not want a direct result. I would like to understand why I think in a logic way: "well, if at 3 m the picture occupies 3 m^2, then, at 5 m will occupy 5 m^2". I mean, what is wrong with this? – John Mar 19 '12 at 19:54
Suppose for example that the picture is $3$ metres wide and $1$ metre high when projected from $3$ metres. Then when you project from $5$ metres the picture will be $(5/3)(3)$ metres wide and $(5/3)(1)$ metres high. Just think in terms of similar triangles, with one vertex at the lens. So the new area is $[(5/3)(3)][(5/3)(1)]$, which is $25/9$ times the original area. – André Nicolas Mar 19 '12 at 20:01

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