Question

In Exercises \(58-60\), use the following information. In \(1803,\) the Louisiana Purchase added \(8.28 \times 10^{5}\) square miles to the United States. The cost of this land was \(\$ 15\) million. In \(1853,\) the Gadsden Purchase added \(2.94 \times 10^{4}\) square miles, and the cost was \(\$ 10\) million. Find the average cost of a square mile for each of the purchases.

Short answer

The average cost per square mile for the Louisiana Purchase was approximately $18.12, and for the Gadsden Purchase, it was approximately $340.14.

Step-by-step solution

Calculate Average Cost of Louisiana Purchase

For the Louisiana purchase in 1803, the total cost was $15 million and the area acquired was \(8.28 \times 10^{5}\) square miles. So, the average cost per square mile can be found by dividing the total cost by the total area. \[ \frac{{\$15,000,000}}{{8.28 \times 10^{5}}\]

Calculate Average Cost of Gadsden Purchase

For the Gadsden purchase in 1853, the total cost was $10 million and the area purchased was \(2.94 \times 10^{4}\) square miles. The average cost per square mile is obtained by dividing the total cost by the total area. \[ \frac{{\$10,000,000}}{{2.94 \times 10^{4}}\]

Solve for Average Cost of Each Purchase

Solve both of the above calculations to find the average cost per square mile for each purchase.

Key concepts

Louisiana Purchase

The Louisiana Purchase, completed in 1803, is a landmark event in American history. It involved the acquisition of approximately 828,000 square miles of territory from France at a cost of \(15 million. This purchase essentially doubled the size of the United States at that time, extending the nation's boundary to the Rocky Mountains and setting the stage for westward expansion.

When discussing the Louisiana Purchase in the context of average cost calculations, we use the total purchase cost and the total land area to determine the cost per square mile. The calculation suggests that each square mile cost about \)18.07, which reflects a remarkably economical rate for such a significant addition to U.S. territory.

Gadsden Purchase

The Gadsden Purchase was a smaller but strategically important acquisition made in 1853, where the U.S. bought approximately 29,670 square miles of land from Mexico for \(10 million. This land purchase secured land for a southern transcontinental railroad and settled the boundaries of the contiguous United States as we know them today.

To calculate the average cost of land during this historical purchase, we take the purchase price and divide it by the number of square miles acquired. For the Gadsden Purchase, the average cost per square mile turns out to be significantly higher at around \)337.46, reflecting the strategic value attributed to the land at that time.

Square Miles Calculation

Calculating square miles is essential when analyzing land purchases, as it allows comparison of the size of different acquisitions. To find the average cost per square mile, which is a common way to compare the value of land deals, you divide the total cost by the total land area in square miles.

Using this calculation, one can obtain a clear picture of the value and scale of historical land purchases. For example, a large purchase with a low average cost per square mile indicates a more economical acquisition, while a smaller area with a high cost per square mile suggests a premium was placed on the land, perhaps due to its location or resources.

Historical Land Purchases

Historical land purchases, like the Louisiana and Gadsden Purchases, have shaped nations and their economies. By calculating the average cost per square mile, we learn more than just the cost; we understand the context of the purchase, the perceived value of the land at the time, and its impact on a nation's growth.

These purchases are often driven by both economic and strategic motivations, as land can provide resources, security, and room for expansion. The value ascribed to land can vary greatly depending on accessibility, fertility, location, resources, and geopolitical significance. Knowing how to analyze these purchases through average cost calculations provides insight into how and why nations have expanded their territories throughout history.