Key Concepts: the Principle of “Same Sum, Smaller Difference, Bigger Product”

Imagine you’re a gardener, and you have a straight wall. You also have a piece of wire mesh that is 36 meters long. You want to use this wall as one side and the wire mesh to enclose a rectangular vegetable garden. The question is, how can you arrange this wire mesh to get the maximum area for your garden?

Let’s draw a picture to visualize this:

==================================================== Wall
            |                    |  
      Wire  |        Garden      | Wire
            |____________________|
                     Wire

As you can see, the wire mesh only encloses three sides of the garden. Let’s say the lengths of the sides perpendicular to the wall are $x$ meters, and the length of the side parallel to the wall is $y$ meters. We know that the total length of the wire mesh is $36$ meters, so we can write this as an equation: $2x + y = 36$.

The area $s$ of the garden is given by the product of the lengths of its sides, or $s = x*y$.

Now, here’s a fun fact: when the sum of two numbers is fixed, their product is maximized when the numbers are as close to each other as possible. In other words, “same sum, smaller difference, bigger product”. But wait, you might say, $x + y$ isn’t fixed here! That’s true, but remember, we’re not looking at $x$ and $y$, we’re looking at $2x$ and $y$, and their sum is fixed at $36$.

So, to maximize the area of the garden, we want 2x and y to be as close as possible. In fact, we want them to be equal. So, let’s set $2x = y$. Substituting this into our original equation, we get $2x + 2x = 36$, or $4x = 36$. Solving for $x$, we find that $x = 9$ meters. Substituting $x = 9$ into $2x = y$, we find that $y = 18$ meters.

So, the maximum area of the garden is $s = x*y = 9 * 18 = 162$ square meters.

And there you have it! By using a little bit of geometry and algebra, and a fun principle of math, you’ve figured out how to maximize your gardening space. Now, go plant some veggies!