Introduction
Rational functions are typically introduced in 11th grade and explored further in 12th grade and college. While they may seem abstract, they play a crucial role in real-life applications, such as designing efficient electrical batteries, optimizing airplane parts, or determining the most sustainable package shapes. In this article, we’ll dive into the last example to see how rational functions solve real-world problems.
What is a sustainable package?
Almost every product—whether solid, liquid, or granular—requires proper packaging. The shape and dimensions of the package vary depending on the product, but it’s always important to use materials economically and sustainably. Both goals mean using as little material as possible. Let’s explore a simple example of how rational functions help find the most efficient package dimensions.
Rational function for sustainable package: a case study
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Designing a box-shaped package
Imagine a packaging company needs to design a shipping box with a square base and a volume of 600 cubic inches. What dimensions should it have? Should the base be 5×5 inches, 10×10 inches, or something else? For example, the three boxes shown below all have a volume of 600 cubic inches, but which one uses the least material?
To find out, we need to compare their surface areas, as explained in the next section.
Calculating surface area
The first box has a surface area of 530 square inches, the second has 440 square inches, and the third has 610 square inches. Clearly, the second box uses the least material for the same volume, but is it the most efficient? To answer this, let’s write a general formula for the surface area:
\[A = 4 \times l \times h + 2l^2\]
Introducing Rational Function
The formula above is a mathematical function of two variables, l and h. However, we can simplify it to a one-variable function.
We know the volume of the box must be 600 cubic inches, so:
\[600 = l^2 \times h\]
Solving for h gives:
\[h = \frac{600}{l^2}\]
Now, substitute this into the surface area formula:
\[A = 4 \times l \times \left( \frac{600}{l^2} \right) + 2l^2 = 2l^2 + \frac{2400}{l}\]
This is now a rational function with one variable, showing how the surface area depends on the length of the box’s base.
Graphing and searching for an optimal value
Let’s plot this function on a graph.
This graph shows how the surface area changes based on the length of the box’s base (while keeping the volume constant at 600 in³). The three boxes from earlier can also be found on this graph (blue dots where l=5, 10 and 15 inches).
You’ve probably guessed that the lowest point on the graph represents the minimum surface area. Here, the smallest surface area is 427 square inches, achieved when the box’s base is 8.43 x 8.43 inches.
Calculating the height of the box
Now that we know the optimal base dimensions, let’s calculate the height:
\[h = \frac{600}{(8.43)^2} \approx 8.43 \text{ inches}\]
As you can see, the height is also 8.43 inches. This means the most efficient shape—with the best volume-to-surface ratio—is a cube.
Rational function as representation of changing values (Conclusion)
Our rational function showed how the surface area of a box with a volume of 600 in³ changes based on the dimensions of its square base:
- Before the minimum point, the box was taller than it was wide.
- At the minimum point, the height and base became equal, forming a cube.
- Beyond that point, the box became wider as the base grew larger than the height.
This example demonstrates how real-world problems — like optimizing package dimensions — can be described mathematically and visualized. Such visualizations help us understand how variables interact and allow us to find optimal solutions.
Of course, different products require different packaging—a pizza box and a bottle box won’t look the same. But this example highlights a general principle: rational functions are powerful tools for solving real-world optimization problems.
Further reading
Interested in more real-world problems involving mathematical functions? Scroll down to the list of tags below and click on the one that interests you most.
If you’re curious about other ways math contributes to sustainability, check out these DARTEF articles: