Problem 52
Question
The Louisiana Purchase. In \(1803,\) the United States negotiated the Louisiana Purchase with France. The country doubled its territory by adding \(827,000\) square miles of land for \(\$ 15\) million. If the land appreciated at the rate of \(6 \%\) each year, what would one square mile of land be worth in \(2005 ?\)
Step-by-Step Solution
Verified Answer
One square mile of land would be worth \(313,805\) in 2005.
1Step 1: Identify the Total Number of Years
Determine the number of years between 1803 and 2005. This is done by subtracting 1803 from 2005, yielding 202 years.
2Step 2: Calculate the Initial Price per Square Mile
Divide the total cost of the purchase ($15 million) by the number of square miles (827,000) to find the price per square mile in 1803.\[\text{Price per square mile} = \frac{15,000,000}{827,000} \approx 18.14\]
3Step 3: Apply Compound Interest Formula
Use the compound interest formula to find the value in 2005.\(A = P(1 + r)^t\), where \(P\) is the initial value ($18.14), \(r\) is the interest rate (6% or 0.06), and \(t\) is the number of years (202).
4Step 4: Substitute and Compute
Substitute the values into the compound interest formula and compute:\[A = 18.14 \times (1 + 0.06)^{202}\]Calculate \((1 + 0.06)^{202}\), and then multiply by 18.14 to find the value of one square mile in 2005.
5Step 5: Final Computation
After calculating \((1.06)^{202} \approx 17,292.90\), multiply by 18.14to get the value in 2005:\[A \approx 18.14 \times 17,292.90 \approx 313,805\]
Key Concepts
Louisiana PurchaseLand AppreciationPrice per Square MileHistorical Calculations
Louisiana Purchase
The Louisiana Purchase was a monumental event in American history that took place in 1803, marking a significant expansion for the United States. At the time, the country acquired approximately 827,000 square miles of land from France for the bargain price of $15 million. This acquisition effectively doubled the size of the United States, providing vast new territories for exploration and settlement.
This purchase was largely orchestrated by President Thomas Jefferson, who saw it as an opportunity to secure land for future generations and fulfill the nation's "Manifest Destiny". The fortunate timing of this purchase was influenced by France's internal pressures and the Haitian Revolution, which made the sale advantageous for both parties involved.
This purchase was largely orchestrated by President Thomas Jefferson, who saw it as an opportunity to secure land for future generations and fulfill the nation's "Manifest Destiny". The fortunate timing of this purchase was influenced by France's internal pressures and the Haitian Revolution, which made the sale advantageous for both parties involved.
Land Appreciation
Land appreciation refers to the increase in the value of the land over time. For investments like real estate, appreciation is an important factor as it can significantly enhance the asset's value.
In the context of the Louisiana Purchase, the land acquired appreciated at an annual rate of 6%. This means that every year, the value of the land increased by 6% of the previous year's value, following the compound interest principle.
Understanding this growth is crucial, as it highlights why such investments are valuable in the long term. Over the years, the compound nature of appreciation significantly increases the value of the asset—transforming a relatively small initial outlay into a potentially sizeable fortune.
In the context of the Louisiana Purchase, the land acquired appreciated at an annual rate of 6%. This means that every year, the value of the land increased by 6% of the previous year's value, following the compound interest principle.
Understanding this growth is crucial, as it highlights why such investments are valuable in the long term. Over the years, the compound nature of appreciation significantly increases the value of the asset—transforming a relatively small initial outlay into a potentially sizeable fortune.
Price per Square Mile
When the United States agreed to the Louisiana Purchase in 1803, the territory was purchased for \(15 million, covering 827,000 square miles. To find the original price per square mile, one would divide the total cost by the total area of land. Using this calculation:
- Total Price = \)15,000,000
- Total Area = 827,000 square miles
Historical Calculations
In historical contexts, calculations often help us understand the true scale or value of past transactions. The usage of compound interest is an essential element when analyzing historical assets like the Louisiana Purchase.
To find out the worth of the land in 2005, using the compound interest formula \(A = P(1 + r)^t\), where:
To find out the worth of the land in 2005, using the compound interest formula \(A = P(1 + r)^t\), where:
- \(P\) is the initial price: \(18.14
- \(r\) is the rate of appreciation: 6% or 0.06
- \(t\) is the number of years since the purchase: 202
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