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So the problem is:

I have bought a fence 30 meters long and I need to put it around three of my rectangular fields sides. How long should be each of the field sides, to create the biggest possible field size?

My field sides are $a$ and $b$, the perimeter of the three sides is $2a + b = 30$ and the size of the field is $a\times b$. I don't know how to solve this problem, but I think I should use derivatives and get the value of the tangent line to the parabola of the $a$ or $b$ graph. But I'm not sure how to do it.

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From $2a+b=30$, you know that $b=30-2a$. That means that the size of the field can be written in terms of $a$ alone: $S(a) = ab = a(30-2a) = 30a-2a^2.$

Now, the side of the field can be as little as $0$, and as large as $15$ (since you only have 30 meters).

So you want to find the maximum of the function $S(a)$, with $0\leq a \leq 30$. This is an optimization problem, which can be solved by (i) finding the critical points; (ii) comparing the value of the function at the critical points and the end points of the relevant interval.

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Use $2a+b=30$ to get $b=30-2a$. Substitute that back in to the area formula to obtain $f(a)=a(30-2a)=30a-2a^2$. If you want you can differentiate that and find the maximum by finding the zeroes of the derivative.