Problem 13 - Applied Calculus

Question

During the 1980 s, Costa Rica had the highest deforestation rate in the world, at \(2.9 \%\) per year. (This is the rate at which land covered by forests is shrinking.) Assuming the rate continues, what percent of the land in Costa Rica covered by forests in 1980 will be forested in \(2015 ?\)

Step-by-Step Solution

Answer

About 38.3% of Costa Rica's forested land remains in 2015.

Step 1: Identify the Problem

We need to find out what percentage of Costa Rica's forested land remains in 2015, given a 2.9% annual deforestation rate starting from 1980.

Step 2: Determine Time Span

The number of years from 1980 to 2015 is calculated by subtracting 1980 from 2015, which gives us 35 years.

Step 3: Use the Exponential Decay Formula

The formula for exponential decay is \( A = P(1-r)^t \), where \( P \) is the initial amount (100% forest in 1980), \( r \) is the decay rate (2.9% or 0.029), and \( t \) is the time in years (35 years).

Step 4: Substitute the Known Values

Insert the known values into the formula: \( A = 100(1-0.029)^{35} \).

Step 5: Calculate the Result

Compute \( A = 100(0.971)^{35} \). By calculating \( (0.971)^{35} \), we find that it is approximately 0.383. Therefore, \( A = 100 \times 0.383 = 38.3 \).

Step 6: Interpret the Result

The result indicates that approximately 38.3% of the forested land in Costa Rica would remain in 2015.

Key Concepts

Understanding Deforestation Rate

The deforestation rate is the speed at which forested areas are being lost over a certain period of time. It's usually expressed as a percentage, representing the proportion of forested land lost annually. In the context of Costa Rica during the 1980s, a 2.9% deforestation rate meant that each year, 2.9% of the remaining forested land was being cut down or converted to other uses. This doesn't just mean 2.9% of the original forest each year. Rather, it's 2.9% of whatever forest was left from the previous year. If unchecked, such loss can lead to significant reductions in forest cover over time, affecting biodiversity and climate. Understanding this rate helps in calculating how quickly forested areas will shrink if no conservation efforts are implemented.

Exploring the Exponential Decay Formula

The exponential decay formula is a mathematical tool used to model processes like deforestation, where quantities decrease over time. The basic formula is given as: \[A = P(1-r)^t\]where:

• \(A\) is the remaining amount after time \(t\),
• \(P\) is the initial amount (in this case, 100% forested land in 1980),
• \(r\) is the decay rate (like 2.9% written as 0.029), and
• \(t\) is the number of years the process continues (35 years, from 1980 to 2015).

The formula works by reducing the initial amount by a consistent percentage, simulating real-world situations where decreases don't happen linearly but accelerate over time as the remaining amount becomes smaller. To use the formula, you substitute the known values, helping you find out how much remains after a specific time period with a certain rate of decay.

Simplifying Percentage Calculation

Percentage calculation is a fundamental skill that helps in understanding various quantitative problems. In the context of exponential decay, you're not simply subtracting the same amount each year, but recalculating based on what's left after each year. This is why the exponential decay formula is so effective; it accounts for continuous percentage reductions. To calculate a percentage: you multiply the original amount by the percentage (as a decimal). For instance, if we want to find out what 2.9% is as a decimal, we convert it by dividing by 100, giving us 0.029. When you perform operations in the exponential decay formula, you multiply that rate against the remaining forested land each year, which gives you a continually shrinking quantity. By the end of these calculations, you end up with a percentage that represents the portion of the forest that remains, which can then be easily interpreted. Thus, learning these percentage calculations ensures that you can understand and compute results accurately in many real-world applications, like assessing deforestation impacts.