Problem 3 - Algebra and Trigonometry

Question

Squirrel Population A grey squirrel population was in- troduced in a certain county of Great Britain 30 years ago. Biologists observe that the population doubles every 6 years, and now the population is \(100,000\) . (a) What was the initial size of the squirrel population? (b) Estimate the squirrel population 10 years from now. (c) Sketch a graph of the squirrel population.

Step-by-Step Solution

Answer

(a) 3,125 squirrels. (b) 317,480 squirrels. (c) Exponential growth graph.

Step 1: Understand the Problem

We are given that a squirrel population doubles every 6 years and currently there are 100,000 squirrels after 30 years. We need to find the initial population size, estimate the population in 10 years from now, and sketch a graph of the population over time.

Step 2: Determine the Number of Doubling Periods

The population doubles every 6 years. Given that it’s been 30 years, divide 30 by 6 to find out how many times the population has doubled.\[\frac{30}{6} = 5\]This means the population has doubled 5 times.

Step 3: Calculate Initial Population Size

Let the initial squirrel population be denoted as \(P_0\). Each doubling period multiplies the population by a factor of 2. Therefore, if it doubles 5 times, the relationship is:\[P_0 \times 2^5 = 100,000\]Solving for \(P_0\):\[P_0 = \frac{100,000}{2^5} = \frac{100,000}{32} = 3,125\]So, the initial population was 3,125 squirrels.

Step 4: Estimate Population 10 Years from Now

In 10 years, 10/6 = 1.67 doubling periods would occur after 30 years. The population can be calculated by:\[P_{10} = 100,000 \times 2^{\frac{10}{6}}\]Calculate the exponent first:\[2^{\frac{10}{6}} \approx 2^{1.67} \approx 3.1748\]Thus, \[P_{10} \approx 100,000 \times 3.1748 \approx 317,480\]So, the estimated population in 10 years is approximately 317,480 squirrels.

Step 5: Sketch the Graph

Plot a graph to represent the squirrel population over time. On the x-axis, represent time in years from 0 to 40. On the y-axis, plot the population in thousands. - Start the graph at 3,125 on the y-axis at 0 years. - Plot points for each 6-year doubling interval: 6 years (6,250), 12 years (12,500), 18 years (25,000), 24 years (50,000), and 30 years (100,000). - Show the estimate for 40 years (317,480). The curve should show an exponential increase, becoming steeper over time.

Key Concepts

Population Doubling

Population doubling is a concept often seen in exponential growth scenarios where, in certain intervals, a population size doubles itself. In other words, it grows by a factor of two. This is a key aspect of exponential growth and helps to predict how a population can rapidly expand over time.

• For the squirrels, their population doubles every 6 years. It shows how dynamic and rapidly changing biological populations can be.
• Understanding the doubling period is crucial, as it provides a framework to calculate past and future population sizes.

Once established, the doubling period allows biologists, like in this problem, to predict changes over long stretches of time without continuous data collection.

Initial Population Size Calculation

To calculate the initial population size of the squirrels, we need to work backward from the current known population. Based on the exponential growth model:

• The formula we use includes the current population and the number of times the population has doubled since its initial introduction.
• This can be expressed as: \(P_0 \times 2^n = P_{current}\), where \(P_0\) is the initial population size, \(P_{current}\) is the current population, and \(n\) is the number of doubling periods.

For this exercise, we know the population is 100,000 squirrels after 30 years and that it doubles every 6 years, giving us 5 doubling periods. Using the formula:\[\frac{100,000}{2^5} = \frac{100,000}{32} = 3,125\]Thus, the initial population size was 3,125 squirrels. Now you can understand how knowing the doubling period and working backwards helps solve such problems efficiently.

Future Population Estimation

Estimating the future population size involves predicting how the current population will continue to grow. Using the known rate of doubling from the problem:

• We estimate how many more doubling periods will occur in the future timeline.
• Specifically, for the squirrel population, we predict it over 10 additional years.

Given 10 years and the doubling period of 6 years here’s how you calculate future population:Use the formula again: \(P_{future} = P_{current} \times 2^{\frac{t}{d}}\), where \(t\) is the future time span, and \(d\) is the doubling period. For this case:\[P_{10} = 100,000 \times 2^{\frac{10}{6}}\]Calculating: \[2^{\frac{10}{6}} \approx 3.1748\] Resulting in: \[P_{10} = 100,000 \times 3.1748 \approx 317,480\]This tells us that the estimated squirrel population in 10 years will be around 317,480 squirrels, illustrating how exponential growth significantly alters population sizes over short time frames.