Problem understanding proportions with exponents

[ksinelli]

I just started reading my physics book again, and one of the very first things it talks about is proportionality. I understand the concept that two things are proportional if one gets multiplied by a certain factor and the other one has to be multiplied by the same factor. For instance, if John makes $10 per hour and works 3 hours, he makes $30. But if he works 6 hours we know that he makes $60 because the money he earns is proportional to the time he spends working.

However, it's when exponents are introduced into the proportion that I get confused. My physics book states that

"the area of a circle is proportional to the square of the radius (A=\pir^{2}, so A \propto r^{2}). The area must increase by the same factor as the radius squared, so if the radius doubles, the area increases by a factor of 2^{2}=4"

I don't understand this. Why are they talking about the radius doubling when the proportionality deals with the radius squared? Shouldn't it be if the radius squared doubles, then the area also doubles? The way they are saying it is like... if the radius doubles, then the area quadruples.

I guess I *kind of* understand what they are doing. They are taking the exponent from the proportionality and using it on the multiplier, but why?

Why isn't it just "if the radius squared is multiplied by a factor, then the area is multiplied by the same factor" ?

 

[hgfalling]

We don't usually talk about the change in the radius squared. I mean, if I offer you an upgrade of a pizza from a 10" pizza to a 12" pizza, I don't say, "I'll give you a pizza with a radius squared of 36 in² instead of a radius squared of 25 in²."

Or maybe the variation will be as the square root of something (as happens often, especially in statistics). So now say the standard deviation of something varies as the square root of the number of things sampled. Or it varies as the inverse of something.

All in all, it is easier for everyone if we just talk about the change of something in absolute units, and then we convert for the type of proportionality in a particular system.

 

[HallsofIvy]

Look at a specific example- The area of a square 2 m on a side if 4 square meters. Now put identical square on the left and above that square and put a fourth square between those two. Those four squares now make a large square that has twice the length and width but four times the area.

 

[ksinelli]

Look at a specific example- The area of a square 2 m on a side if 4 square meters. Now put identical square on the left and above that square and put a fourth square between those two. Those four squares now make a large square that has twice the length and width but four times the area.

i don't deny that the proportionality is true. I've substituted numbers for r and the answer comes out just as the book says. I do appreciate your help though.

We don't usually talk about the change in the radius squared. I mean, if I offer you an upgrade of a pizza from a 10" pizza to a 12" pizza, I don't say, "I'll give you a pizza with a radius squared of 36 in² instead of a radius squared of 25 in²."

All in all, it is easier for everyone if we just talk about the change of something in absolute units, and then we convert for the type of proportionality in a particular system.

I've thought about this also, and it seems to be in the right direction of understanding for me, but my problem with this is that if you double the radius first and then square that answer, it's a different answer than squaring the radius first and then doubling that answer. Which seems to be the logical way it should be done, because the proportionality states that A \propto r^2. It seems that r^2 should be treated as a single entity and that you shouldn't be able to do anything to just r.

Perhaps I am just not stating clearly enough what it is exactly that has me confused, but I'm not sure how else to explain it.