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You will build a rectangular sheep pen next to a river. There is no need to build a fence along the river, so you only need to build three sides. You have a total of 450 feet of fence to use. Find the dimensions of the pen such that you can enclose the maximum area.

janh callm
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  • What have you tried? Make two variables, one for the length and one for the width. The area is the product. The $450$ gives a relation between them, so plug that in and have an equation for the area as a function of one. Differentiate, set to zero,..... Where is your problem? – Ross Millikan Mar 14 '17 at 04:07
  • He probably cannot use calculus, this is a fairly low-level question. – Robin Aldabanx Mar 14 '17 at 04:09
  • -b/2a works instead though, so I agree – Robin Aldabanx Mar 14 '17 at 04:09

2 Answers2

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The way to solve this is to set $x$ to be the width of the rectangle and $y$ to be the length.

Therefore: $2x+y=450$, because of the restriction on the perimeter. $xy=$ the area.

Let us solve for $y$ in terms of $x$.

$y=450-2x$

Therefore, our area is $x(450-2x)$ or $-2x^2+450x$. Since this is an upside-down parabola (the negative coefficient on the $x^2$-value), we can use $-\frac{b}{2a}$ to find the maximum.

Our $b$ = 450, and $a=-2$, so we have the maximum area at the $x$-value:

$$x=-\frac{450}{-4}$$ $$x=112.5$$ By our first definition: $y=450-2x$

$$y=450-2 \cdot 112.5$$ $$y=225$$

Our maximum dimensions are length $225$ and width $112.5$, with area being $25312.5$.

Robin Aldabanx
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We differentiate function of area wrt a and equating it with 0 SO that we can calc. a for which area will be max.

user41111
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