Problem 64 - Algebra 2

Question

RAILROADS For Exercises \(64-66,\) use the following information. The First Transcontinental Railroad was built by two companies. The Central Pacific began building eastward from Sacramento, California, while the Union Pacific built westward from Omaha, Nebraska. The two lines met at Promontory, Utah, in \(1869,\) approximately 6 years after construction began. The Central Pacific Company laid an average of 9.6 miles of track per month. Together the two companies laid a total of 1775 miles of track. Determine the average number of miles of track laid per month by the Union Pacific Company.

Step-by-Step Solution

Answer

Union Pacific laid approximately 15.05 miles of track per month.

Step 1: Define Variables

Let the average number of miles of track laid per month by the Union Pacific Company be \( x \). The Central Pacific Company's average was given as 9.6 miles per month.

Step 2: Utilize Total Track Laid

The total track laid by both companies is 1775 miles. If we let the tracks laid by Central Pacific be \( c \) and by Union Pacific be \( u \), then:\[u = 1775 - c\]

Step 3: Calculate Time Function

Both companies worked for approximately 72 months (6 years). So:\[c = 9.6 \times 72\]and\[u = x \times 72\]

Step 4: Calculate Central Pacific's Contribution

Calculate \( c \):\[c = 9.6 \times 72 = 691.2\]

Step 5: Solve for Union Pacific's Contribution

Substitute \( c \) into the total equation:\[u = 1775 - 691.2 = 1083.8\]

Step 6: Find Union Pacific's Monthly Average

Use \( u = x \times 72 \) to find \( x \):\[x = \frac{1083.8}{72} \approx 15.05\]

Step 7: Solution Summary

The average number of miles of track laid per month by the Union Pacific Company is approximately 15.05 miles.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different variables through operations like addition, subtraction, multiplication, and division. They are crucial in solving various types of mathematical problems, including those involving rates of work. In the context of the railroad problem, algebraic equations help us determine unknown values by using known information. Given two companies laying track, we know their combined effort totaled 1775 miles. Here, the equation represents the relationship between the total miles laid and the miles laid individually by each company. To find the average rate of work for the Union Pacific Company, algebraic equations allow us to express the total miles laid by both companies and solve for the unknown variable denoting the Union Pacific's average work rate per month. This systematic way of forming and solving equations is an essential skill in algebra, aiding complex decision-making in real-world scenarios.

Transcontinental Railroad

The First Transcontinental Railroad was a remarkable feat of engineering and logistics. Built in the 1860s, it connected the eastern United States with the Pacific coast, substantially cutting travel time across the country. The Union Pacific Railroad built westward, starting from Omaha, Nebraska, while the Central Pacific Railroad started from Sacramento, California, heading eastward. They met at Promontory, Utah in 1869. This monumental project spanned a total of 1775 miles, providing a seamless rail link across the American continent. This project not only transformed transportation but also showcased incredible teamwork and coordination between different companies. Studying the transcontinental railroad offers insight into project management, engineering challenges, and the impact of infrastructure on societal progress. The careful planning and division of work among multiple groups are similar to problem-solving in algebra, where different parts work in unison to achieve a final, useful result.

Average Speed

Average speed is a fundamental concept that describes how fast something moves over a given distance and time period. It's calculated by dividing the total distance covered by the time taken to cover that distance. In the railroad problem, the Central Pacific Company's average speed of laying track was 9.6 miles per month; this provides a measure of their rate of work. Similarly, finding the average speed for the Union Pacific Company involves determining how many miles they laid on average per month over the same time period. Understanding average speed helps us compare the efficiency of different processes. For the railroad project, which involves complex logistical challenges, knowing average speeds allows companies to plan more effectively and make necessary adjustments to ensure project timelines are met.

Distance-Time Relationship

The relationship between distance and time is a core principle of physics and mathematics that describes how they interact through the concept of speed or rate. This principle is vital when dealing with problems involving travel or the laying of tracks in a railroad project.In the exercise, the distance-time relationship is depicted by the formulae used to calculate the track length laid over a set period. The total distance the railways covered was 1775 miles, and this work took around 72 months to complete.By analyzing how distance and time relate, we can use simple arithmetic and algebra to find unknown quantities, such as the average rate at which the Union Pacific laid tracks. This involves using the formula: \[ \text{Distance} = \text{Rate} \times \text{Time} \]which lets us solve various aspects of the problem efficiently. This helps in making predictions and strategizing future work plans based on past performance, a practice common in many fields beyond railroad construction.