Wikipedia:Reference desk/Archives/Mathematics/2009 July 21

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July 21

sizes of round objects

My local pizza shop has for the same price, four 9 inch pizzas, three 12 inch pizzas or two 15 inch pizzas. which is the most pizza and how do you work it out? we also hate crusts? which has the most crust and hoe do you work it out? —Preceding unsigned comment added by Payneham (talk • contribs) 05:00, 21 July 2009 (UTC)

The formula to find an area of a circle is πr². Assuming 9-inch, 12-inch, and 15-inch refer to the pizzas' diameters, then r (the radius of a pizza), would be half of these values, so 4.5 inches, 6 inches, and 7.5 inches, respectively. So the area of each (rounding to the nearest square inch) would be:

π(4.5)² = 64 in²

π(6)² = 113 in²

π(7.5)² = 177 in²

Now multiply by the quantity of each pizza to find the total area of each pizza "combo": 4 x 64 = 256 in², 3 x 113 = 339 in², 2 x 177 = 354 in². Therefore the two 15-inch pizzas have the most pizza for the money. However how much crust depends almost entirely, I suppose, on the kind of pizza it is. Ginogrz (talk) 05:18, 21 July 2009 (UTC)

The multiplication by π is not needed in order to answer the question, and omitting it makes the answers exact:

${\begin{aligned}4\cdot {\frac {\pi }{4}}\cdot 9^{2}={\frac {\pi }{4}}\cdot 324\\\\3\cdot {\frac {\pi }{4}}\cdot 12^{2}={\frac {\pi }{4}}\cdot 432\\\\2\cdot {\frac {\pi }{4}}\cdot 15^{2}={\frac {\pi }{4}}\cdot 450\end{aligned}}$

The ratio of areas is 324 : 432 : 450.

Never round until the last step. Michael Hardy (talk) 12:43, 21 July 2009 (UTC)

Assuming the crust has the same thickness on all the pizza sizes and is pretty thin compared to the overall size of the pizzas, the amount of crust on each pizza is going to be pretty much proportional to the length around the outside of the pizza, which is the circumference. Circumference equals π times the diameter, so the first combo has 4 x 9in x π = 36π in of crust, the second combo has 3 x 12in x π = 36π in, and the third combo has 2 x 15in x π = 30π in. So the two 15in pizzas also have the least crust. Rckrone (talk) 06:24, 21 July 2009 (UTC)

To minimize the arithmetic to what can be done without computer: Note that the diameters, 9 12 15 inches, are simply 3 4 5 times some length unit, 3 inches. So the areas per pizza are 9 16 25 times some common area unit. The area of pizza per price is then (4×9, 3×16, 2×25) = 36 48 50, and the third option is obviously the one with most area of pizza. The amount of crust goes as the circumference, which is as (4×3, 3×4, 2×5) = 12 12 10. The third option is the one with least crust. Bo Jacoby (talk) 08:03, 21 July 2009 (UTC).

I want to point out that Jacoby's solution makes use of proportions, which certainly simplifies the arithmetic a lot, as well as removing the need to use the mathematical constant π to calculate the areas, which is ugly and can only be done approximately. To write out what he did more explicitly, we have the formula:

$A=\pi r^{2}$

Which translates to a proportionality:

$A\propto r^{2}=({\frac {d}{2}})^{2}={\frac {d^{2}}{4}}\propto d^{2}$

The conversion from radius to diameter above shows that:

$A\propto d^{2}$

And similarly, the formula for a circumference:

$C=\pi d$

Directly leads to:

$C\propto d$

Since they preserve order, the proportional quantities can be compared exactly, without the worry of rounding errors. Of course, you may still want to know the areas and circumferences of the pizzas, in which case you have to approximate using π. --COVIZAPIBETEFOKY (talk) 11:53, 21 July 2009 (UTC)

Changing someone else homework

If say, someone has done their mathematical homework.

If I being a naughty person and went to changed all the positive numbers into negative numbers by changing the sign and vice versa. Will the homework still be correct? 122.107.207.98 (talk) 13:01, 21 July 2009 (UTC)

No. If he wrote $2\cdot 2=4$, changing it to $(-2)\cdot (-2)=-4$ will no longer be correct.

To put it differently, $x\mapsto -x$ is not an automorphism of the field of real numbers. Compare this to $x\mapsto {\bar {x}}$, which is an automorphism of the field of complex numbers. -- Meni Rosenfeld (talk) 13:20, 21 July 2009 (UTC)

Estimate of Perron-Frobenius Theorem

In Perron–Frobenius theorem, we read that an estimate is $\min _{i}\sum _{j}a_{ij}\leq r\leq \max _{i}\sum _{j}a_{ij}$. While this is not too hard to understand, I need a citeable source for this statement in the non-negative case and so far failed to find one. Does anyone know any on the web (especially in Google-books, which I have been browsing through without success yet)? --KnightMove (talk) 17:43, 21 July 2009 (UTC)