Problem 90 - Elementary Algebra

Question

The proposed Nabucco pipeline will transport oil from the eastern coast of Turkey to Austria. The pipeline travels \(3900 \mathrm{~km}\) through Bulgaria, Romania, and Hungary and will cost \(7.9\) billion euros. Find the cost of the pipeline per mile in U.S. dollars. ( \(1 \mathrm{mi}=1.6 \mathrm{~km} ; 1\) euro \(=\) \(1.3145\) U.S. dollars.) Round to the nearest hundredth of a million. (Sources: www.nabucco-pipeline.com; www.ecb .int, Jan. 27, 2012)

Step-by-Step Solution

Answer

4.26 million U.S. dollars per mile

Step 1 - Convert kilometers to miles

First, convert the total length of the pipeline from kilometers to miles. Use the conversion factor \(1\ \text{mi} = 1.6\ \text{km} \). To find the total length in miles, divide the total length in kilometers by the conversion factor: \[ 3900\text{ km} \div 1.6 = 2437.5 \ \text{miles} \]

Step 2 - Calculate the cost per kilometer in euros

Next, determine the cost per kilometer by dividing the total cost in euros by the total length in kilometers: \[ \text{Cost per km} = \frac{7.9 \ \text{billion euros}}{3900 \text{ km}} \ To simplify the calculation, convert 7.9 billion to a number : 7.9 \cdot 10^9 euros} \ = 2025641.03 \ \text{euros/km}\]

Step 3 - Convert the cost per kilometer from euros to U.S. dollars

Use the given exchange rate to convert euros to U.S. dollars: \[ \text{1 euro} = 1.3145 \text{ U.S. dollars}\] To find the cost per kilometer in U.S. dollars: \[ \text{Cost per km in U.S. dollars} = 2025641.03 euros/km \times 1.3145 = 2661895.79 USD/km\]

Step 4 - Determine the cost per mile in U.S. dollars

Finally, calculate the cost per mile by multiplying the cost per kilometer in U.S. dollars by the conversion factor \( 1\text{ mile} = 1.6 \text{ km} \). \[ \text{Cost per mile in U.S. dollars} = 2661895.79 \text{ USD/km} \times 1.6 = 4259033.26\ \text{USD/mile} \]

Step 5 - Round to the nearest hundredth of a million

Round the final cost per mile to the nearest hundredth of a million:4259033.26 U.S. dollars can be approximated to 4.26 million U.S. dollars per mile.

Key Concepts

Unit Conversion

Unit conversion is crucial in solving problems where different measurement systems are involved. Here, we convert the pipeline length from kilometers to miles. Knowing that 1 mile equals 1.6 kilometers, we divide the total length by this conversion factor:
\ \( \frac{3900 \text{ km}}{1.6} = 2437.5 \text{ miles} \)
This step ensures that all subsequent calculations are performed using the same unit, making the problem easier to manage.

• Step-by-Step: Identify the units needing conversion
• Choose an appropriate conversion factor: Here, it's 1 mile = 1.6 km
• Perform the division: Total km divided by 1.6 to get the miles

Currency Conversion

Converting currencies involves using a given exchange rate to switch from one monetary unit to another. In this exercise, we move from euros to U.S. dollars. Utilizing the exchange rate of 1 euro = 1.3145 US dollars, we have:
\ \( 2025641.03 \text{ euros/km} \times 1.3145 = 2661895.79 \text{ USD/km} \)
Currency conversions are essential in international projects to provide a clearer financial picture, using a consistent currency unit.

• Identify necessary conversions: Euros to USD
• Use the exchange rate: 1 euro = 1.3145 US dollars
• Execute the multiplication: Euros/km converted by multiplying with the given rate

Rounding Numbers

Rounding numbers simplifies the presentation of results, making them more comprehensible. Here, we round the pipeline cost per mile to the nearest hundredth of a million. The computed cost is approximately 4.259 million U.S. dollars, which is rounded to:
\ \( 4.26 \text{ million USD} \)
This step is particularly useful in financial reporting and engineering estimates where precision to some significant figures is enough.

• Calculate to required precision: Here, four decimal places
• Apply rounding rules: If the next digit is ≥5, round up
• Present the rounded number: Gives a clear, readable figure

Pipeline Cost

Understanding pipeline costs involves various factors including length, material, labor, and even geographical considerations. Here, we start with the total project cost in euros and narrow it down to the cost per mile. Initially:
\ \( \frac{7.9 \text{ billion euros}}{3900 \text{ km}} = 2025641.03 \text{ euros/km} \)
Converted to U.S. dollars:
\ \( 2025641.03 \text{ euros/km} \times 1.3145 \text{ USD/euro} = 2661895.79 \text{ USD/km} \)
Finally, we find the cost per mile:
\ \( 2661895.79 \text{ USD/km} \times 1.6 = 4259033.26 \text{ USD/mile} \)

• Identify total project costs: 7.9 billion euros
• Breakdown per unit distance: Euros/km
• Convert currency: to USD
• Compute cost per mile: multiplying USD/km by conversion factor