# How do you find the dimensions of the rectangle of greatest area whose perimeter is 20 cm?

Feb 7, 2017

$5 \text{cm" xx 5 "cm}$

#### Explanation:

Let the length of one side of the rectangle be $x$ $\text{cm}$.

Then the opposite side is also of length $x$ $\text{cm}$, while the two other sides are of length:

$\frac{20 - 2 x}{2} = 10 - x$ $\text{cm}$

The area of the rectangle is:

$x \left(10 - x\right) = 10 x - {x}^{2} = 25 - 25 + 10 x - {x}^{2} = 25 - {\left(x - 5\right)}^{2}$ ${\text{cm}}^{2}$

This attains its maximum, $25$, when $x = 5$

Hence the rectangle of maximum area is a $5 \text{cm" xx 5 "cm}$ square.