Area of Goalbox vs. Area of Penalty Box (rationals)

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The discussion focuses on calculating the size difference between the soccer penalty box and the goal box using their respective area formulas. The area of the goal box is given as (10x^2) / 27, while the penalty box area is (22x^2) / 9. To find how many times larger the penalty box is compared to the goal box, the ratio is set up as (22x^2) / 9 divided by (10x^2) / 27. This simplifies to (22x^2 / 9) * (27 / 10x^2), allowing for the cancellation of x^2 terms. The final calculation will yield the ratio indicating the size difference between the two areas.
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Homework Statement



Area of goal box (soccer field) is described with (10x^2) / 27 and area of penalty box is described with (22x^2) / 9 . Determine how many times greater in size the penalty box is compared to the goal box.

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The Attempt at a Solution



I created the areas of the boxes based from dimensions, and simplified to what I have given (assume they are correct). Just don't get how to figure out that size difference (which is terrible since I just completed this question on the Fermat math contest for Waterloo earlier today, no problem, but I get thrown off in even easier math?)
 
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read the question again. you don't even have to know the areas... it is simply asking for a ratio between the two
 
well ya... Simple ratio is (10x^2)/27 : (22x^2)/9 ... sooo.. would I then put that into fraction form (not sure which on top vs bottom for this situation), then simplify as far as possible...?
 
"Determine how many times greater in size the penalty box is compared to the goal box."
You want "penalty box" over "goal box".

(22x^2) / 9 over (10x^2) / 27 which is the same as (22x^2)/9 times 27/(10x^2).
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

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