Problem 82 - Brief Applied Calculus

Question

Many chemical and refining companies use "the rule of point six" to estimate the cost of new equipment. According to this rule, if a piece of equipment (such as a storage tank) originally cost \(C\) dollars, then the cost of similar equipment that is \(x\) times as large will be approximately \(x^{0.6} C\) dollars. For example, if the original equipment cost \(C\) dollars, then new equipment with twice the capacity of the old equipment \((x=2)\) would cost \(2^{0.6} \mathrm{C}=1.516 \mathrm{C}\) dollars-that is, about \(1.5\) times as much. Therefore, to increase capacity by \(100 \%\) costs only about \(50 \%\) more. \({ }^{*}\) Use the rule of 6 to find how costs change if a company wants to triple \((x=3)\) its capacity.

Step-by-Step Solution

Answer

The cost triples capacity and increases by around 93.3%.

Step 1: Understand the Problem Statement

We need to find how costs change when the capacity of equipment is tripled (i.e., when \(x = 3\)). We will use the rule of point six formula \(x^{0.6} C\), where \(C\) is the original cost and \(x\) is the multiplier of capacity.

Step 2: Apply the Rule of Point Six

Substitute \(x = 3\) into the equation \(x^{0.6} C\). The formula becomes \(3^{0.6} C\).

Step 3: Calculate \(3^{0.6}\)

Using a calculator, compute \(3^{0.6}\). This gives us approximately 1.933.

Step 4: Estimate New Equipment Cost

Multiply the result from Step 3 by \(C\) to get the cost of the new equipment. Thus, the new cost will be approximately \(1.933C\).

Step 5: Interpret the Result

The new piece of equipment with triple capacity will cost about 1.933 times the cost of the original equipment. This means the cost increases by around 93.3%.

Key Concepts

Cost Estimation

Cost estimation is an essential aspect of financial planning in any industry, especially in sectors such as chemical and refining, where equipment costs can be significant. "The rule of point six", used widely in these industries, is a practical tool for cost estimation. It simplifies the complex process of predicting costs for new, larger equipment based on existing costs.
This rule states that the cost of equipment scaled by a factor of \( x \) can be estimated by \( x^{0.6} C \), where \( C \) is the original cost. The exponent 0.6 is key as it suggests that costs don't increase linearly with scale. As equipment size increases, the cost rises at a slower rate, reflecting economies of scale. This estimation method helps industries budget efficiently, as it gives them a quick way to predict financial outlay without detailed analyses of every component.
In practice, if a company knows the cost of a smaller equipment piece, it can easily project larger-scale requirements. This estimate, while not exact, is usually sufficient for high-level budget planning, helping businesses navigate cost-effective scaling strategies.

Scaling Laws

Scaling laws describe how different factors change with scale. These laws are prevalent in various scientific and engineering fields. In cost estimation, for instance, the rule of point six serves as a scaling law by predicting how equipment costs grow as size increases.
What makes scaling laws like this one useful is their ability to reveal patterns in growth. They demonstrate non-linear relationships, where growth in one dimension affects others unpredictably, but consistently.

• This particular law shows that costs increase at a rate slower than the proportional increase in size, which suggests economies of scale and indicates more efficient use of resources.
• These laws help clarify what to expect when making adjustments or expansions, providing a predictable formula.

Scaling laws thus serve as a powerful tool for engineers and financial planners, offering a framework for expecting changes based on size adjustments. Understanding these laws aids in constructing cost-effective and efficient scaling strategies.

Exponential Functions

Exponential functions are a mathematical concept commonly used to describe growth processes. When it comes to cost estimation, an exponential function helps us understand how costs might grow with size. In the rule of point six, we see an exponential component \( x^{0.6} \) which shows that cost growth is not straightforward.
This function shows that a doubling or tripling of capacity does not simply double or triple the cost. Instead, costs grow at a diminishing rate relative to capacity increase.
For example:

• Doubling capacity (\( x = 2 \)) means multiplying the cost by \( 2^{0.6} = 1.516 \), an increase of about 51.6%.
• Tripling capacity (\( x = 3 \)) results in a cost increase by \( 3^{0.6} \), which is about 1.933, or 93.3% more.

This diminishing growth captured by the exponential function can help industries plan their resources more effectively by providing realistic expectation models. Exponential functions thus translate abstract mathematical ideas into practical applications.