Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Land value In 1867 the United States purchased Alaska from Russia for \(\$ 7,200,000\). There is 586,400 square miles of land in Alaska. Assuming that the value of the land increases continuously at \(3 \%\) per year and that land can be purchased at an equivalent price, determine the price of 1 acre in the year 2010 . (One square mile is equivalent to 640 acres.)

Step-by-Step Solution

Answer

The price of 1 acre in 2010 is approximately $1.39.

Step 1: Understand the Problem

We need to calculate the price of 1 acre of land in Alaska in 2010, given continuous growth in land value from 1867. We will use the continuous compound interest formula to find the total land value in 2010 and then determine the price per acre.

Step 2: Use Continuous Compound Interest Formula

The continuous compound interest formula is given by: \[ P(t) = P_0 e^{rt} \]Where:- \( P_0 = 7,200,000 \) (initial land value)- \( r = 0.03 \) (rate of growth)- \( t = 2010 - 1867 = 143 \) yearsSubstitute the values into the formula:\[ P(143) = 7,200,000 \, e^{0.03 \, \times \, 143} \]

Step 3: Calculate the Total Land Value in 2010

Compute \( e^{0.03 \, \times \, 143} \): \[ e^{4.29} \approx 72.43 \]Therefore, the total land value in 2010 is:\[ P(143) = 7,200,000 \, \times \, 72.43 \approx 521,496,000 \]

Step 4: Convert Land Area From Square Miles to Acres

Given that one square mile is equivalent to 640 acres, we need to find out how many acres are in Alaska:\[ 586,400 \, \text{square miles} \, \times \, 640 \, \text{acres/square mile} = 375,296,000 \, \text{acres} \]

Step 5: Determine Price per Acre in 2010

Divide the total land value from Step 3 by the total number of acres:\[ \text{Price per acre} = \frac{521,496,000}{375,296,000} \approx 1.39 \]Thus, the price per acre of land in Alaska in 2010 is approximately \( \$ 1.39 \).

Key Concepts

Land Value Over Time

When discussing land value, it's essential to appreciate how it can change over time. The value of land can be affected by a variety of factors including economic conditions, interest rates, and even government regulations. In the context of Alaska, the land was initially purchased for $7,200,000 back in 1867. But land value doesn't stay static; it typically grows over time due to factors such as development and demand.
For this particular situation, we are assuming that the land value increases continuously at a rate of 3% per year. This continuous growth rate is a simplified model for understanding how investments can grow over a period of time. Therefore, we need to use a specific mathematical formula to calculate how much the land would be worth in 2010. This helps us understand not just historical land value but also the potential for appreciation over time.

Acreage Conversion Essentials

Understanding acreage conversion is a vital skill when dealing with large plots of land. In the United States, land area is often measured in both square miles and acres. If you know the area of a large land parcel in square miles, converting to acres is straightforward yet essential for land valuation or agricultural purposes.
To convert square miles to acres, remember that 1 square mile equals 640 acres. This constant is crucial in scenarios where you're dealing with massive amounts of land area. For instance, Alaska comprises 586,400 square miles, a relatively large number that can be difficult to conceptualize. By converting this into acres, which results in 375,296,000 acres, the scale becomes more comprehensible for those needing detailed land-use planning or valuation.

Exponential Growth and Land Valuation

Exponential growth is a fundamental concept in understanding how investments or values can multiply over time. It differs from simple interest in that it considers compounding at every possible instant, theoretically leading to significantly larger outcomes.
The continuous compound interest formula, expressed as \( P(t) = P_0 e^{rt} \), is an essential tool here. In this formula, \( P_0 \) is the initial amount (or land value), \( r \) is the growth rate, \( t \) is the time in years, and \( e \) is the base of the natural logarithm. Applying this formula helps determine the future value of the land given continuous growth. In our Alaska example, this continuous compounding significantly increases the land value from its original purchase price to over 500 million dollars by 2010.
Understanding exponential growth not only applies to land value but also to a broad range of financial applications, making it an invaluable skill.