Problem 81 - 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 quadruple \((x=4)\) its capacity.

Step-by-Step Solution

Answer

The cost will increase to approximately 2.297 times the original cost if capacity is quadrupled.

Step 1: Understand the Question

We are given a cost estimation rule for equipment based on the relative size of its capacity. Our task is to find the cost increase when the capacity is increased by four times (i.e., when \(x=4\)).

Step 2: Apply the Rule of Point Six

According to the rule, the cost \(C_x\) for equipment \(x\) times larger than the original is \(x^{0.6} \cdot C\). We substitute \(x=4\) into this equation.

Step 3: Calculate \(4^{0.6}\)

Calculate the value of \(4^{0.6}\) to find out how much the cost increases for a quadruple capacity. Using a calculator, \(4^{0.6} \approx 2.297\).

Step 4: Interpret the Result

Multiplying the original cost \(C\) by \(4^{0.6} \approx 2.297\) gives us the estimated cost of the new equipment: \(2.297 \cdot C\). This means the cost is approximately 2.297 times the original cost.

Key Concepts

Cost Estimation

Cost estimation plays a crucial role in budgeting and financial planning, particularly in industries like chemical and refining. When companies plan to purchase new equipment, they need a reliable method to predict expenses. Instead of relying solely on past prices, businesses often use cost estimation rules, which help them gauge potential costs based on various factors, like equipment size or capacity.

• Estimation is crucial for setting budgets and forecasting financial needs.
• It ensures companies allocate appropriate funds, avoiding overspending or underspending.

For instance, if a company estimates that a new tank will cost about $100,000, and that estimate is off by 50%, it could lead to significant financial complications. Thus, understanding and applying these estimation rules accurately is essential for smooth business operations.

Rule of Point Six

"The rule of point six" is a widely used guideline in industries that handle equipment scaling. It simplifies the estimation of costs when equipment size changes. This rule states that if original equipment costs \( C \) dollars, equipment \( x \) times larger would cost approximately \( x^{0.6} C \) dollars.

• It provides a straightforward method to calculate cost changes due to capacity changes.
• The rule suggests that costs do not scale linearly with capacity.

For example, if a company needs equipment that is twice as large as the original, it would not simply cost double. Instead, according to this rule, it would cost about 1.5 times more (since \(2^{0.6} \approx 1.516\)). Applying the rule helps businesses make quick but informed decisions about investments in new equipment.

Capacity Increase Calculation

Capacity increase calculation using the rule of point six involves a sequence of straightforward steps. Suppose a company aims to quadruple its equipment capacity. Using the given rule, we need to determine how the costs will change.

• First, understand the relationship: the cost adjustment factor is given by \( x^{0.6} \).
• Then, apply \( x = 4 \) for quadrupling capacity.

When we compute \( 4^{0.6} \), we find it approximately equals 2.297. Thus, by multiplying this factor by the original cost \( C \), the new cost estimate becomes \( 2.297 \, C \). This reveals that quadrupling the capacity increases the cost to about 2.297 times the initial expense. This calculation helps companies understand the financial implications of scaling their equipment and allows them to plan accordingly.