Problem 44 - Calculus

Question

In November 2010 , a "tiger summit" was held in St. Petersburg, Russia. \(^{28}\) In \(1900,\) there were 100,000 wild tigers worldwide; in 2010 the number was 3200 . (a) Assuming the tiger population has decreased exponentially, find a formula for \(f(t),\) the number of wild tigers \(t\) years since 1900 (b) Between 2000 and 2010 , the number of wild tigers decreased by \(40 \%\). Is this percentage larger or smaller than the decrease in the tiger population predicted by your answer to part (a)?

Step-by-Step Solution

Answer

The 37.8% decrease predicted is smaller than the actual 40% decrease.

Step 1: Understand Exponential Decay Formula

The number of tigers is assumed to decrease exponentially. The general formula for exponential decay is \(f(t) = f_0 \cdot e^{-kt}\), where \(f_0\) is the initial quantity, \(k\) is the decay constant, and \(t\) is the time.

Step 2: Identify Initial Data Points

In this problem, \(f_0 = 100,000\) (the number of tigers in 1900), and \(f(t) = 3200\) when \(t = 110\) (in 2010).

Step 3: Solve for the Decay Constant k

Substitute the known values into the decay formula: \(3200 = 100,000 \cdot e^{-110k}\). Solving for \(k\) gives:\[3200 = 100,000 \cdot e^{-110k} \\frac{3200}{100,000} = e^{-110k} \\ln\left(\frac{32}{1000}\right) = -110k \k \approx \frac{\ln\left(\frac{32}{1000}\right)}{-110}\]

Step 4: Write the Exponential Function

Using the calculated value of \(k\), the function representing the tiger population over time can be written as:\[f(t) = 100,000 \cdot e^{-0.0466t}\] (assuming \(k \approx 0.0466\))

Step 5: Calculate 2000 Population Using the Model

For 2000, \(t = 100\), substitute into the function:\[ f(100) = 100,000 \cdot e^{-0.0466 \times 100} \approx 5,148\]

Step 6: Determine Percentage Decrease From 2000 to 2010

Initial population in 2000 is approximately \(5,148\), and in 2010 it is \(3,200\). Find the percentage decrease:\[\text{Percentage Decrease} = \frac{5148 - 3200}{5148} \times 100\% \approx 37.8\%\]

Step 7: Compare with Actual Given Decrease

The actual given decrease in tiger population from 2000 to 2010 is \(40\%\). Our exponential model predicts a \(37.8\%\) decrease, which is slightly smaller.

Key Concepts

Exponential Growth and Decay

Exponential growth and decay are natural processes observed in various contexts, such as population dynamics and radioactive decay. In the case of decay, the quantity decreases over time. This concept is expressed using the exponential decay formula:

• \(f(t) = f_0 \cdot e^{-kt}\)

where:

• \(f(t)\) is the amount at time \(t\),
• \(f_0\) is the initial amount,
• \(k\) is the decay constant, and
• \(t\) is the time elapsed since the start.

The use of the natural base \(e\) (approximately 2.718) is crucial, as it ensures continuous decay proportional to the current amount. In the context of the exercise above, the number of tigers declines exponentially, starting from an initial count of 100,000 in 1900 to a much reduced number in 2010.

Decay Constant

The decay constant, \(k\), is a critical component of the exponential decay formula. It determines the rate at which the population or quantity decreases. To find \(k\), you rearrange the exponential decay formula and solve using known values. For instance, when the initial population of tigers was 100,000, and it reduced to 3,200 over 110 years, we substitute these values to solve for \(k\):

• \[3200 = 100,000 \cdot e^{-110k}\]

By taking the natural logarithm,

• \[\ln\left(\frac{3200}{100,000}\right) = -110k\]
• You find \(k = \frac{\ln\left(\frac{32}{1000}\right)}{-110} \approx 0.0466\).

This value helps create the exponential function that accurately models how the tiger population changes over time, using the rate determined by the decay constant.

Population Modeling

Population modeling involves using mathematical equations to represent the changes in a population over time. With exponential decay, we model a decrease using a specific formula that incorporates the decay constant. In the case of the wild tiger population:

• The model is \(f(t) = 100,000 \cdot e^{-0.0466t}\).

This equation predicts the population size at any given time \(t\) since 1900. For example, to find the population in the year 2000, you set \(t = 100\) and calculate:

• \[f(100) = 100,000 \cdot e^{-0.0466 \times 100} \approx 5148\]

Population modeling is essential for understanding trends, making forecasts, and planning conservation efforts. In this case, the model helps to compare the actual 40% decrease from 2000 to 2010 against the model's predicted 37.8% decrease, offering insights into the pattern of tiger population decline.