Problem 90 - Applied Calculus

Question

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 triple \((x=3)\) its capacity.

Step-by-Step Solution

Answer

If capacity is tripled, costs increase by approximately 93.3%.

Step 1: Understand the Rule

The rule of 0.6 states that the cost of new equipment is approximately given by the expression \( x^{0.6} C \), where \( x \) is the factor by which the equipment size increases and \( C \) is the original cost.

Step 2: Determine the Cost Factor for Tripling Capacity

In this scenario, we need to triple the capacity, meaning \( x = 3 \). Using the rule, the cost of the new equipment will be \( 3^{0.6} C \).

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

To find out how much the cost increases, we need to calculate \( 3^{0.6} \). This is approximately equal to 1.933.

Step 4: Apply the Cost Increase

The cost for the new equipment that triples the capacity will be \( 1.933C \), meaning the cost increases by a factor of approximately 1.933 times the original cost.

Key Concepts

Cost Estimation

When it comes to planning for new equipment, cost estimation is key to business success. In industries like chemical and refining, companies need reliable methods to predict costs when scaling their operations. The "rule of point six" is an insightful method used to estimate these costs effectively. According to this rule, if you know the cost of a particular piece of equipment, you can estimate the cost of a larger or smaller version using the formula \( x^{0.6} \times C \). Here, \( x \) represents the size increase factor, and \( C \) is the original cost.

This approximation helps businesses decide whether scaling their equipment is financially viable. For example, if a company wants to double its equipment size, multiplying the original cost by \( 2^{0.6} \) gives an accurate cost estimate. Understanding this simple multiplicative relationship allows companies to budget accurately and avoid unexpected expenses.

Scaling Calculations

Scaling calculations are essential when a business plans to expand or adjust its operations. The rule of point six provides a simplified yet effective way to perform these calculations. In this context, scaling refers to changing the size of equipment such as storage tanks, reactors, or other industrial components.

To compute the cost of new equipment when scaling, businesses use the exponent 0.6 in the formula \( x^{0.6} \times C \). This exponent captures how costs increase in a non-linear fashion compared to the equipment's size increase. For instance, increasing equipment size by a factor of 3 involves raising 3 to the power of 0.6, resulting in approximately 1.933. Therefore, scaling up does not linearly increase costs, which is crucial for decision-making.

• Linear scaling would mean tripling the size triples the cost. However, by using the 0.6 exponent, costs rise at a slower rate.
• This method effectively balances between the economy of scale and added complexity or expense due to larger size.

Exponential Growth

Exponential growth is a concept that describes processes where the rate of growth is proportional to the current value, causing the growth rate to accelerate over time. In business, it's vital to understand exponential growth due to its implications on scaling and budgeting.

In the context of the rule of point six, exponential growth helps explain why costs do not increase linearly with equipment size. The formula \( x^{0.6} \times C \) effectively models how capacity increases involve varying levels of cost increase. Here, the exponent 0.6 efficiently translates the concept of diminishing returns; as you scale equipment, although you require more resources, the increase in cost per unit of capacity decreases.

• This is important for businesses not only for equipment but also for processes, investments, and technologies that undergo similar scaling principles.
• Understanding exponential growth allows management to predict changes and budget for future projects intelligently.

Recognizing these nuances helps companies plan long-term investments and strategize effectively in growing competitive markets.