Question

Bald eagle population Since DDT was banned and the Endangered Species Act was passed in 1973 , the number of bald eagles in the United States has increased dramatically (see figure). In the lower 48 states, the number of breeding pairs of bald eagles increased at a nearly linear rate from 1875 pairs in 1986 to 6471 pairs in 2000. a. Use the data points for 1986 and 2000 to find a linear function $p$ that models the number of breeding pairs from 1986 to $2000(0\le t\le 14)$ b. Using the function in part (a), approximately how many breeding pairs were in the lower 48 states in $1995?$

Short answer

Answer: Approximately 4827 breeding pairs of bald eagles were estimated to be in the lower 48 states in 1995.

Step-by-step solution

Define the given data points and variables

We are given two data points: - In 1986, there were 1875 breeding pairs of bald eagles (t = 0, p = 1875). - In 2000, there were 6471 breeding pairs of bald eagles (t = 14, p = 6471). We need to find a linear function p(t), with t being the number of years since 1986 (0 ≤ t ≤ 14).

Find the slope of the linear function

To find the slope (m) of the function, we will use the formula: $m=\frac{y_2-y_1}{x_2-x_1}$ In this case, x represents the years (t) and y represents the number of breeding pairs (p): $m=\frac{6471-1875}{14-0}=\frac{4596}{14}=328$ So, the slope of the linear function is 328.

Find the y-intercept of the linear function

To find the y-intercept (b) of the function, we can use the slope-intercept form of a linear function: $p=mt+b$ We have the slope (m = 328) and one data point (t = 0, p = 1875). Plug these values into the formula: $1875=328\cdot 0+b$ So, the y-intercept of the linear function is b = 1875.

Write the linear function

Now that we have the slope (m = 328) and the y-intercept (b = 1875), we can write the linear function: $p(t)=328t+1875$ This function models the number of breeding pairs of bald eagles in the lower 48 states from 1986 to 2000 (0 ≤ t ≤ 14).

Estimate the population in 1995

To estimate the number of breeding pairs in 1995, we need to find the value of the function at t = 9 (as 1995 is 9 years after 1986): $p(9)=328\cdot 9+1875=2952+1875=4827$ Using the linear function, we estimate that there were approximately 4827 breeding pairs of bald eagles in the lower 48 states in 1995.

Key concepts

Slope Calculation

Understanding the concept of 'slope' is crucial for analyzing how a function behaves, especially for linear functions like the one modeling the bald eagle population. Slope essentially measures the rate of change between two points on a line. If you imagine the line as a path, the slope tells us how steep this path is.

To find the slope of a linear function, we use the formula:

• $m=\frac{y_2-y_1}{x_2-x_1}$

Here, $(x_1,y_1)$ and $(x_2,y_2)$ are two distinct points on the line. In our exercise, these points are given by years since 1986 and the number of eagle pairs in those years.

Using this formula gives us a slope of 328, which tells us that every year, starting from 1986, about 328 new pairs of bald eagles were being added. This increased rate is a direct reflection of improved conditions for their population growth.

Y-intercept

The y-intercept of a line is where the line crosses the y-axis, which is when $x=0$. For our problem, the y-axis represents the number of bald eagle pairs and $x=t$, where $t$ is the number of years since 1986.

To find the y-intercept, we use one of our known data points. Since the number of breeding pairs in 1986 (when $t=0$) was 1875, the y-intercept $b$ is simply 1875.

So, the equation of the line can be written as:

• $p(t)=328t+1875$

The y-intercept here represents the starting number of breeding pairs in 1986. This kind of interpretation is crucial when we're analyzing growth trends, as it sets the baseline from which change is measured.

Population Estimation

Population estimation using a linear function involves computing values within a given range by substituting specific timeframes into the function. In our scenario, we want to predict the number of bald eagle pairs in 1995.

Since 1995 is 9 years after 1986, we set $t=9$ and substitute it into the linear function:

• $p(9)=328\times 9+1875$

After doing the math, we find around 4827. This estimate helps assess how the bald eagle population has changed over time due to environmental protections and conservation efforts.

Such predictions make it easier for conservationists and policy-makers to track progress and adjust strategies as needed. Linear functions not only estimate current situations but also can predict future trends if conditions remain consistent.