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| Math Central | Quandaries & Queries |
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Question from Lyndsay, a teacher: A rectangle is to be constructed having the greatest possible area and a perimeter of 50 cm. (a) If one of the sides of the rectangle measures 'x' cm, find a formula for calculating the area of the rectangle as a function of 'x'. (b) Determine the dimensions of the rectangle for which it has the greatest area possible. What is the maximum area? |
Hi Lyndsay,
Suppose the other side has length $y$ cm then the perimeter is $x + x + y + y = 2x + 2y$ cm. But you know this is 50 cm so $2x + 2y = 50 \mbox{ or } x + y = 25.$ Thus $y = 25 - x.$
The area, $A$ of a rectangle is the length times the width and hence $A = x \times y$ or
\[A = x(25 - x).\]
There are a couple of ways to approach part (b). If you know some calculus you can treat part (b) as a max-min problem. Otherwise you can use the fact that the maximum or minimum of the quadratic function $a x^2 + b x + c$ is at the vertex.
Penny
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