Estimating The Drilling Cost For An 80-Meter Deep Hole

July 6, 2025 by Scholario Team 55 views

Calculating the Cost of Drilling an 80-Meter Deep Hole

Drilling deep into the earth requires significant resources, especially when dealing with increasing costs for each meter bored. This article delves into the mathematical calculation of estimating the total cost of drilling an 80-meter deep hole, given that the cost for the first meter is SSP 30 and increases for each subsequent meter. This exploration involves understanding arithmetic progressions and their application in real-world scenarios. The principles discussed here are applicable not only to drilling costs but also to various other situations involving sequentially increasing expenses.

Understanding the Problem

To accurately estimate the cost, we need to break down the problem into manageable parts. The core concept here is an arithmetic progression, a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, the cost increases by a fixed amount (SSP 1) for each meter drilled. This makes it an arithmetic progression problem, which can be solved using specific formulas.

Key Elements of the Problem

Depth of the hole: 80 meters
Cost of the first meter: SSP 30
Increase in cost per meter: SSP 1

These key elements form the foundation of our calculation. We can define the sequence of costs as follows:

Meter 1: SSP 30
Meter 2: SSP 31
Meter 3: SSP 32
And so on...

The challenge is to find the total cost of drilling all 80 meters. Manually adding up each cost would be time-consuming and prone to errors. Therefore, we need a more efficient method.

Applying Arithmetic Progression Formulas

Arithmetic progressions have well-defined formulas that allow us to calculate the sum of a sequence. Two primary formulas are essential for solving this problem:

The nth term of an arithmetic progression: ${ a_n = a_1 + (n - 1)d }$ $a_{n} = a_{1} + (n - 1) d$ Where:
- ${ a_n }$ is the nth term
- ${ a_1 }$ is the first term
- ${ n }$ is the number of terms
- ${ d }$ is the common difference
The sum of the first n terms of an arithmetic progression: ${ S_n = \frac{n}{2} [2a_1 + (n - 1)d] }$ $S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) d]$ Or, alternatively: ${ S_n = \frac{n}{2} (a_1 + a_n) }$ $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ Where:
- ${ S_n }$ is the sum of the first n terms

These formulas are crucial for determining the total drilling cost. The first formula helps us find the cost of the 80th meter, and the second formula allows us to calculate the total cost for all 80 meters.

Step-by-Step Calculation

Let's apply these formulas to our problem:

Find the cost of the 80th meter ( ${ a_{80} }$ ):
- ${ a_1 = 30 }$
- ${ n = 80 }$
- ${ d = 1 }$ ${ a_{80} = 30 + (80 - 1) imes 1 }$ ${ a_{80} = 30 + 79 }$ ${ a_{80} = 109 }$ So, the cost of drilling the 80th meter is SSP 109.
Calculate the total cost of drilling 80 meters ( ${ S_{80} }$ ): Using the formula ${ S_n = \frac{n}{2} (a_1 + a_n) }$ $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ :
- ${ n = 80 }$
- ${ a_1 = 30 }$
- ${ a_{80} = 109 }$ ${ S_{80} = \frac{80}{2} (30 + 109) }$ ${ S_{80} = 40 imes 139 }$ ${ S_{80} = 5560 }$ Therefore, the total cost of drilling the 80-meter hole is SSP 5560.

This calculation provides a clear and concise estimate of the drilling cost. The use of arithmetic progression formulas simplifies what could be a cumbersome manual calculation.

Practical Implications and Considerations

Understanding the mathematical aspect of the problem is only part of the equation. In practical terms, several other factors could influence the actual cost of drilling. These considerations include:

Additional Costs

Equipment and labor: The cost of machinery and the wages of the drilling crew are significant expenses.
Maintenance and repairs: Drilling equipment requires regular maintenance, and unexpected repairs can add to the cost.
Transportation: Moving equipment and materials to and from the drilling site incurs transportation costs.
Permits and regulatory fees: Depending on the location, permits and regulatory fees may be required.
Geological factors: The type of soil and rock can affect the drilling speed and, consequently, the cost.

Risk Management

Unexpected geological conditions: Encountering hard rock or other obstructions can increase drilling time and costs.
Equipment failure: Malfunctions can halt operations and necessitate costly repairs.
Weather conditions: Inclement weather can delay drilling, leading to increased expenses.

Considering these factors provides a more realistic view of the overall cost. While the arithmetic progression calculation gives us a base estimate, incorporating these additional variables is crucial for effective budgeting and project management.

Real-World Applications

The principles of arithmetic progressions extend beyond drilling costs. They are applicable in various real-world scenarios, such as:

Financial Planning

Savings and investments: Calculating the future value of regular deposits or investments with a fixed interest rate involves arithmetic progressions.
Loan repayments: Understanding the total interest paid on a loan with fixed installments is another application.

Business and Operations

Inventory management: Estimating the total cost of holding inventory with increasing storage fees can be done using arithmetic progressions.
Production planning: Calculating the cumulative output of a manufacturing process with a consistent increase in production rate.

Engineering and Construction

Construction costs: Estimating the total cost of a project with increasing material or labor expenses.
Infrastructure development: Planning the budget for projects where costs escalate over time due to inflation or other factors.

By recognizing the patterns of arithmetic progressions, individuals and organizations can make more informed decisions in various aspects of life and work.

Conclusion

Estimating the cost of drilling an 80-meter deep hole, with an increasing cost per meter, is a practical application of arithmetic progressions. The mathematical formulas provide a reliable method for calculating the total cost, as demonstrated by our calculation of SSP 5560. However, it is essential to remember that this is just an estimate. Real-world factors such as equipment costs, geological conditions, and unforeseen risks can significantly impact the final expense.

By understanding the core mathematical principles and considering the practical implications, stakeholders can develop more accurate budgets and manage drilling projects effectively. Moreover, the applications of arithmetic progressions extend far beyond this specific scenario, making it a valuable tool in financial planning, business operations, and various other fields. This exploration highlights the importance of mathematical literacy in addressing everyday challenges and making informed decisions.

Therefore, while the cost of drilling the 80-meter hole is estimated at SSP 5560, a comprehensive approach that incorporates both mathematical precision and practical considerations is necessary for successful project execution. This ensures that the estimate the cost is not just a theoretical exercise but a realistic and actionable assessment.