Every year, endangered species experience population decline and are pushed closer to extinction. This decline in population is mostly caused by human destruction of these species' natural habitat. The Western Lowland Gorilla is an example of an endangered species. In 2022, there are 360,000 Western Lowland Gorillas remaining, with an annual population decline of 2.7%.
a. Use what you've learned in this unit to model the population of Western Lowland Gorillas after 5, 10 and 20 years. Let y equal the population of the gorillas and x represent the number of years since 2022. Show your work.
b. Use the information calculated in step A to create a table showing the Gorilla population after 5, 10 and 20 years.
c. Explain why the table shows exponential decay. Summarize how scientists can use exponential decay to predict population changes in endangered species. Summarize your answer in 1-2 paragraphs.
a. y = 360,000(0.973^x)
b.
Years since 2022 | Gorilla Population
------------------|------------------
5 | 343,945
10 | 288,945
20 | 166,945
c. The table shows exponential decay because the population of the gorillas is decreasing by a consistent rate each year. Exponential decay is a mathematical model that can be used to predict population changes in endangered species. This model is based on the idea that the population of a species will decrease by a certain percentage each year. This percentage is determined by the rate of population decline, which is usually caused by human destruction of the species' natural habitat. By using this model, scientists can accurately predict the population of an endangered species in the future and take steps to protect it.
Can I copy this?!?!
a. To model the population of Western Lowland Gorillas after 5, 10, and 20 years, we can use the formula for exponential decay:
y = P(1 - r)^x
where:
y = population after x years
P = initial population
r = annual population decline rate (as a decimal)
x = number of years since 2022
Given:
P = 360,000
r = 2.7% = 0.027
After 5 years:
x = 5
y = 360,000(1 - 0.027)^5
≈ 360,000(0.973)^5
≈ 360,000(0.88739)
≈ 319,660
After 10 years:
x = 10
y = 360,000(1 - 0.027)^10
≈ 360,000(0.973)^10
≈ 360,000(0.85664)
≈ 308,391
After 20 years:
x = 20
y = 360,000(1 - 0.027)^20
≈ 360,000(0.973)^20
≈ 360,000(0.78145)
≈ 281,322
b. The table showing the Gorilla population after 5, 10, and 20 years is as follows:
| Years | Population |
|-------|------------|
| 5 | 319,660 |
| 10 | 308,391 |
| 20 | 281,322 |
c. The table shows exponential decay because the population is decreasing over time, following a pattern where each year's decline is based on a percentage of the previous year's population. As the years increase, the population decreases at a decreasing rate. This is seen in the calculations where the population after 5 years is higher than after 10 years, which is in turn higher than after 20 years.
Scientists can use exponential decay to predict population changes in endangered species by analyzing the initial population, the annual decline rate, and the number of years. With this information, they can estimate future population levels and determine how close a species is to extinction. By understanding the rate of decline and the factors contributing to it, scientists can implement conservation efforts and make informed decisions to protect and help recover endangered species.
a. To model the population of Western Lowland Gorillas after a certain number of years, we can use the formula for exponential decay:
y = P(1 - r)^x
Where:
- y represents the population of gorillas after x years
- P represents the initial population (360,000 gorillas in this case)
- r represents the annual population decline rate as a decimal (2.7% or 0.027)
- x represents the number of years since 2022.
For 5 years:
y = 360,000(1 - 0.027)^5
For 10 years:
y = 360,000(1 - 0.027)^10
For 20 years:
y = 360,000(1 - 0.027)^20
You can calculate these equations to get the population of Western Lowland Gorillas after 5, 10, and 20 years.
b. To create a table showing the Gorilla population after 5, 10, and 20 years, you can substitute the values of x into the equation and calculate the resulting population.
| Years | Gorilla Population |
|-------|------------------|
| 5 | Calculate using the equation y = 360,000(1 - 0.027)^5 |
| 10 | Calculate using the equation y = 360,000(1 - 0.027)^10 |
| 20 | Calculate using the equation y = 360,000(1 - 0.027)^20 |
c. The table shows exponential decay because the population of Western Lowland Gorillas is decreasing over time at a constant rate. Exponential decay occurs when a quantity decreases by a fixed percentage over fixed intervals of time. In this case, the population decline rate of 2.7% per year causes the population to decrease exponentially.
Scientists can use exponential decay to predict population changes in endangered species by studying historical data and calculating the population decline rate. By understanding the rate at which a population is declining, scientists can project future population numbers. This information is crucial for conservation efforts as it helps identify the urgency and scale of actions required to protect and recover endangered species.
Additionally, exponential decay models can be used to estimate the time it might take for a species to become extinct if the current decline rate continues. This allows scientists and conservationists to prioritize conservation efforts and implement strategies to mitigate habitat destruction and other factors contributing to population decline. Overall, exponential decay provides a quantitative framework for understanding and managing endangered species populations.