Celine stands before a dilemma: two cereal boxes, each promising a delightful breakfast experience, but with subtly different dimensions and volumes. The choice isn't simply about aesthetics; it's a mathematical puzzle wrapped in a colorful, crunchy exterior. Which box holds more cereal? The answer, as we'll discover, hinges on a careful consideration of polynomial expressions and the constraints imposed by the width of the boxes.
Introducing the Contenders:
Our two contenders are Box 1 and Box 2. Their volumes are expressed as polynomial functions of a variable, 'x', which we can interpret as a scaling factor related to the dimensions of the boxes. Box 1 boasts a volume of 3x⁶ cubic units, while Box 2 offers a slightly more complex volume of 4x⁶ - x⁴ cubic units. The key to Celine's decision lies in understanding how these volumes change with different values of 'x', particularly when we introduce the condition that the width of the boxes must be greater than 1.
Understanding Polynomial Functions:
Before diving into the comparison, let's briefly review the nature of polynomial functions. These functions are characterized by terms involving variables raised to non-negative integer powers, multiplied by coefficients. In our case, both volumes are sixth-degree polynomials, meaning the highest power of 'x' is 6. This signifies that the volume changes significantly as 'x' increases. The coefficients (3 and 4 for the x⁶ terms) determine the rate of this change.
The crucial difference lies in Box 2's volume: 4x⁶ - x⁴. The subtraction of x⁴ introduces a complexity that requires careful analysis. While the 4x⁶ term dominates for larger values of 'x', the -x⁴ term subtracts from the overall volume, especially when 'x' is relatively small. This means the relative volumes of the boxes are not constant but depend heavily on the value of 'x'.
Analyzing the Volumes for x > 1:
Celine's decision is further constrained by the condition that the width of the cereal boxes must be greater than 1. This condition directly impacts the value of 'x', linking it to the physical dimensions of the boxes. While we don't know the exact relationship between 'x' and the width, we can reason that a larger 'x' generally corresponds to a larger width, given that 'x' is a scaling factor.
Let's analyze the volumes for various values of 'x' greater than 1:
* x = 1.1:
* Volume of Box 1: 3(1.1)⁶ ≈ 4.13
* Volume of Box 2: 4(1.1)⁶ - (1.1)⁴ ≈ 4.87 - 1.46 ≈ 3.41
In this case, Box 1 holds more cereal.
* x = 2:
* Volume of Box 1: 3(2)⁶ = 192
* Volume of Box 2: 4(2)⁶ - (2)⁴ = 256 - 16 = 240
Here, Box 2 holds more cereal.
* x = 3:
* Volume of Box 1: 3(3)⁶ = 1458
* Volume of Box 2: 4(3)⁶ - (3)⁴ = 2916 - 81 = 2835
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