How do you find the maximum area of a rectangle with perimeter of 40 ft?

Feb 9, 2016

Your gut feeling may tell you it's a square, with sides 10 ft, or a total area of 100 sqft.

Explanation:

You can divert from this and see if the area gets any larger, or you can use the mathematical way:
If the length =$x$ and the width =$y$ then the perimeter $P = 40$
$P = 2 x + 2 y = 40 \to x + y = 20 \to y = 20 - x$

As for the area $A$:
$A = x \cdot y = x \cdot \left(20 - x\right) = 20 x - {x}^{2}$
And we have to find an extreme for that:
We can do this by setting the derivative to $= 0$
$A ' = 20 - 2 x = 0 \to x = 10 \to y = 10$
Just as we thought in the first place.
graph{20x-x^2 [-131.6, 135.4, -8.4, 125.1]}