Question

Writing: Suppose that your company makes plastic trays and is planning new ones with dimensions, including thickness of the material, double those now being made. Your company president is convinced that they will need only twice as much plastic as the older version. "Twice the size, twice the plastic," he proclaims, and no one is willing to challenge him. Your job is to make a presentation to the president where you tactfully point out that he is wrong and where you explain that the new trays will require eight times as much plastic. Write your presentation.

Short answer

Doubling the dimensions of the plastic trays results in an eightfold increase in volume due to the mathematical principle that volume scales with the cube of the length scale factor, which in this case is 2 (2^3 = 8). Therefore, the new trays will require eight times as much plastic.

Step-by-step solution

Understanding the Problem

Identify the dimensions of the old trays and understand what doubling the dimensions means. Doubling each dimension (length, width, and thickness) of a rectangular prism like a tray magnifies its volume in three dimensions.

Explaining Volume Scaling

Explain the concept of volume and how it scales with dimension. For a rectangular prism (or tray), the volume is calculated by the product of its length, width, and thickness. Doubling each dimension leads to an eightfold increase in volume, since Volume Scale Factor = (2 * Length) * (2 * Width) * (2 * Thickness) = 2^3 * Original Volume.

Addressing the Misconception

Address the president's misconception by showing the mathematical calculation. Use a sample calculation with real numbers (if available) or hypothetical ones to show that when you double the dimensions, you are not doubling the volume; you are increasing it by a factor of eight.

Proving with a Mathematical Example

If the old tray has dimensions with length = L, width = W, and thickness = T, then the volume is L*W*T. The new tray will have dimensions 2L, 2W, and 2T, resulting in a volume of 2L * 2W * 2T = 8 * (L * W * T). This mathematical example shows the volume is multiplied by eight.

Explaining the Impact on Material Usage

Discuss that since the trays are made of plastic, the amount of plastic needed is directly proportional to the volume of the tray. Therefore, if the new trays have eight times the volume, they will require eight times as much plastic.

Conclusion and Recommendation

Conclude by summarizing the explanation and deliver a recommendation for planning materials procurement or redesigning the new trays to optimize the use of plastic, considering the actual increase in material needs.

Key concepts

Dimensional Analysis

In the realm of mathematics and physics, dimensional analysis is a tool we use to understand how different physical quantities are related to one another. It involves the study of units and dimensions which are used to describe phenomena in the natural world. For instance, when we talk about dimensions with regard to a physical object like a tray, we're referring to its length, width, and height—the basic parameters that define its size.

When it comes to scaling objects, it's critical to perform dimensional analysis to predict how a change in size will affect other properties. For the trays mentioned in the exercise, doubling each dimension doesn't simply double the amount of material required; it actually increases it more significantly because volume is a three-dimensional attribute. Each dimension increases the overall size of the object exponentially, which means that if you double the length, width, and height, the volume increases by a factor of eight, as it involves three dimensions (2×2×2).

In practice, this is vital in industries like manufacturing where precision in material estimation is crucial for cost efficiency and production planning. By neglecting the rules of dimensional analysis, as the company president did, one could underestimate or overestimate the resources needed, leading to financial and resource allocation issues.

Proportional Relationships

Understanding proportional relationships is essential for scaling objects and predicting the impact on their properties. Proportional relationships indicate how quantities change with respect to one another; they are often represented by the equation 'y = kx', where 'y' and 'x' are variables and 'k' is the constant of proportionality.

In the context of our trays, when dimensions are scaled up or down, volume changes proportionally to the cube of the scaling factor. This is because volume is calculated by multiplying the three dimensions together. If you increase each dimension by a factor of two, the volume will increase by a factor of two cubed, which equals eight, reflecting a direct proportional relationship where the volume is directly proportional to the cube of the scaling factor.

However, the president's assumption was based on a linear proportional relationship, suggesting that doubling the size would double the volume. This misunderstanding can lead to underestimating materials and costs. In businesses, recognizing the type of proportional relationship that exists between dimensions and volume is key for accurate predictions and decision-making.

Volume Calculation

Volume calculation is a fundamental concept in geometry that involves determining the amount of space an object occupies. For many shapes, especially in a manufacturing context like the creation of plastic trays, we often deal with rectangular prisms. The volume of a rectangular prism is found by multiplying its length by its width by its height, expressed as 'Volume = Length × Width × Height'.

Doubling the dimensions of a tray means multiplying its length, width, and height each by two. As a consequence, the volume doesn't just double, but scales by two raised to the power of three, due to the three dimensions involved. To put it mathematically, if the original volume is 'V', then the new volume 'V_new' after scaling would be 'V_new = 2^3 × V' or, in layman’s terms, eight times the original volume.

This fundamental misstep in the president’s reasoning is a clear example of the importance of correct volume calculation. Not only does it affect how much material is needed, but it also has broader implications on cost estimation, storage requirements, and transportation logistics in the business sphere. Such calculations aren't just academic; they have real-world impacts on efficiency and profitability.