Math CentralQuandaries & Queries


Question from Lexie:

suppose you want to make a rectangular garden with the perimeter of 24 meters.
What's the greatest the area could be and what are the dimensions?


Hi Lexie,

Suppose that the length of the garden in $L$ meters and the width is $W$ meters. Since the perimeter is the distance all the way around the garden the perimeter $P$ is given by

\[P = 2L + 2W \mbox{ meters}\]

and the area $A$ is given by

\[A = L \times W \mbox{ square meters.}\]

You know that the perimeter is 24 meters we have

\[2L + 2W = 24 \mbox{ meters.}\]

solve this equation for $L$ and substitute into the equation for the area $A.$ This gives an equation for $A$ in terms of $W.$

What you do at this point depends on what you know. If you are studying calculus you can use calculus to find the value of $W$ that maximizes $A.$ Or you can notice that the equation for $A$ in terms of $W$ is a quadratic in $W$ so its graph is a parabola and you can use properties of the parabola to determine the maximum value of $A$ and the value of $W$ that gives this maximum.

Write back if you need more assistance,


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