Problem 89 - Applied Calculus

Question

\(89-90 .\) BUSINESS: The Rule of .6 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 \(x^{06} \mathrm{C}\) 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 quadruple \((x=4)\) its capacity. "Although the rule of .6 is only a rough "rule of thumb," it can be somewhat justified on the basis that the equipment of such industries consists mainly of containers, and the cost of a container depends on its surface area (square units), which increases more slowly than its capacity (cubic units)

Step-by-Step Solution

Answer

The cost increases by about 2.297 times when quadrupling the capacity.

Step 1: Understanding the Problem

We need to apply the rule of .6 to determine how much it costs when a company wants to quadruple its equipment capacity. Given that quadrupling the capacity means that the scaling factor \(x\) is 4, we can use the formula in the problem, i.e., the cost will be approximately \(x^{0.6}C\) dollars.

Step 2: Setting Up the Equations

The original cost of the equipment is denoted by \(C\). We need to find the new cost when \(x = 4\). Therefore, using the given rule, the new cost would be \(4^{0.6} \times C\).

Step 3: Calculating 4^0.6

Compute \(4^{0.6}\) to find out how much the scaling factor affects the cost. Evaluating \(4^{0.6}\) using a calculator gives approximately 2.297.

Step 4: Calculating the New Cost

Multiply the original cost \(C\) by \(4^{0.6}\) to find the cost of the new equipment. This gives us \(2.297C\). Thus, the new equipment will cost about 2.297 times the original cost.

Key Concepts

Rule of .6

The "rule of .6" is an estimation technique used within industries like chemical engineering to predict the cost of scaling up equipment sizes. It suggests that if you need a piece of equipment that is a certain multiple of the original size, the cost doesn’t increase proportionally. Instead, the cost increases more slowly than the size.

This is expressed mathematically as: if the original cost is \( C \), then the cost for equipment \( x \) times larger than the original is approximately \( x^{0.6} \times C \). This provides a cost function where the exponent \( 0.6 \) indicates that costs rise at a decelerating rate when compared to size.

• For instance, doubling the size increases the cost by about 1.5 times (\(2^{0.6} \times C\)), not double.
• This rule helps to quickly estimate the expenses involved in upgrading equipment without detailed calculations.

Understanding this rule can be crucial when budget planning for scaling operations as it encapsulates the economic perspective of larger investments.

Equipment Sizing

When companies decide to increase production capacity, it often involves larger equipment. Equipment sizing becomes an essential aspect to determine and plan for expansions. But how do you decide on the size?

The actual capacity a production unit can handle is directly tied to its size, which in many industries means its volume. If the equipment's volume doubles, its ability to store, process, or produce also doubles. However, the cost does not. Hence, using the rule of .6, companies can make informed predictions about how equipment scaling will financially impact them.

• Sizing up equipment often demands more than just doubling the factory floor space.
• Considerations include logistics, space utilization, labor needs, and potential downtime during installation.

Analyzing these factors allows companies to strategically manage physical scaling without exceeding budget limits.

Cost Scaling

Cost scaling involves determining how costs increase when scaling operations or equipment. The concept brings into play the rule of .6, which shows that costs grow at a less-than-linear rate compared to capacity expansion.

For example, if a company's current equipment handles 100 units and expansion requires equipment for 400 units, according to the rule of .6:

• The scaling factor is 4 (quadrupling).
• The increase in cost is given by \(4^{0.6}\) or about 2.297 times the original cost.

This calculation provides a realistic estimation of new investments, emphasizing that cost scaling is not merely a direct multiplication of current spending but rather a scaled measure reflecting inherent economic efficiencies of larger operations.

Surface Area vs Volume

Understanding the relationship between surface area and volume is key to grasping why the rule of .6 works. Equipment like tanks and silos are predominantly priced based on the surface area of materials involved in construction rather than on the volume they enclose.

• The surface area increases by the square of the scale factor when dimensions increase, while volume increases by the cube.
• This means that as equipment is scaled up, the surface area (and thus the material cost) increases more slowly compared to the volume.

This discrepancy is why the cost curve flattens (due to relying more on surface area than volume), justifying why the rule uses \(0.6\) rather than \(1\) as an exponent. It reflects the practical economic impact where increasing the size yields diminishing cost proportionality.