Type The Correct Answer In Each Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar(s).The Native Bird Population In A City Is Decreasing At A Rate Of $10\%$ Per Year. After 1 Year, The Population Of Native Birds Is
Understanding the Problem
The native bird population in a city is decreasing at a rate of 10% per year. This means that every year, the population of native birds will be 90% of the previous year's population. We are asked to find the population of native birds after 1 year.
Calculating the Remaining Population
To calculate the remaining population after 1 year, we need to multiply the initial population by 0.9 (which is equivalent to 90%). This is because the population decreases by 10% every year, leaving 90% of the previous year's population.
Let's assume the initial population of native birds is 100. After 1 year, the population will be:
100 x 0.9 = 90
So, the population of native birds after 1 year is 90.
Example with a Fraction
Let's consider an example where the initial population of native birds is 100 and the decrease rate is 1/10 (which is equivalent to 10%). After 1 year, the population will be:
100 x (1 - 1/10) = 100 x (9/10) = 90
So, the population of native birds after 1 year is still 90.
General Formula
The general formula to calculate the remaining population after n years is:
P(n) = P(0) x (1 - r)^n
where:
- P(n) is the population after n years
- P(0) is the initial population
- r is the decrease rate (as a decimal)
- n is the number of years
Applying the Formula
Let's apply the formula to our example. We have:
- P(0) = 100 (initial population)
- r = 0.1 (decrease rate as a decimal)
- n = 1 (number of years)
Plugging these values into the formula, we get:
P(1) = 100 x (1 - 0.1)^1 = 100 x (0.9)^1 = 90
So, the population of native birds after 1 year is still 90.
Conclusion
In conclusion, the population of native birds after 1 year is 90, assuming a decrease rate of 10% per year. This can be calculated using the formula P(n) = P(0) x (1 - r)^n, where P(n) is the population after n years, P(0) is the initial population, r is the decrease rate (as a decimal), and n is the number of years.
Practice Problems
- If the native bird population is decreasing at a rate of 15% per year, what will be the population after 2 years, assuming an initial population of 100?
- If the native bird population is decreasing at a rate of 20% per year, what will be the population after 3 years, assuming an initial population of 100?
- If the native bird population is decreasing at a rate of 5% per year, what will be the population after 4 years, assuming an initial population of 100?
Answers
- 75
- 64
- 81
Note: These answers can be calculated using the formula P(n) = P(0) x (1 - r)^n, where P(n) is the population after n years, P(0) is the initial population, r is the decrease rate (as a decimal), and n is the number of years.
Frequently Asked Questions
Q: What is the formula to calculate the remaining population after n years?
A: The formula to calculate the remaining population after n years is:
P(n) = P(0) x (1 - r)^n
where:
- P(n) is the population after n years
- P(0) is the initial population
- r is the decrease rate (as a decimal)
- n is the number of years
Q: How do I calculate the remaining population if the decrease rate is given as a percentage?
A: To calculate the remaining population if the decrease rate is given as a percentage, you need to convert the percentage to a decimal. For example, if the decrease rate is 10%, you can convert it to a decimal by dividing by 100, which gives you 0.1. Then, you can use the formula P(n) = P(0) x (1 - r)^n to calculate the remaining population.
Q: What if the decrease rate is given as a fraction?
A: If the decrease rate is given as a fraction, you can convert it to a decimal by dividing the numerator by the denominator. For example, if the decrease rate is 1/10, you can convert it to a decimal by dividing 1 by 10, which gives you 0.1. Then, you can use the formula P(n) = P(0) x (1 - r)^n to calculate the remaining population.
Q: Can I use the formula to calculate the remaining population if the decrease rate is not constant?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the decrease rate is constant over time. If the decrease rate is not constant, you will need to use a more complex formula or model to calculate the remaining population.
Q: How do I calculate the remaining population if the initial population is not given?
A: If the initial population is not given, you will need to estimate it or use a different formula that does not require the initial population. For example, you can use the formula P(n) = P(0) x (1 - r)^n and substitute P(0) with a known value, such as the population at a previous time period.
Q: Can I use the formula to calculate the remaining population if the population is increasing?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is decreasing. If the population is increasing, you will need to use a different formula that takes into account the increase in population.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the initial population is not given?
A: If the decrease rate is given as a percentage and the initial population is not given, you will need to estimate the initial population or use a different formula that does not require the initial population. For example, you can use the formula P(n) = P(0) x (1 - r)^n and substitute P(0) with a known value, such as the population at a previous time period.
Q: Can I use the formula to calculate the remaining population if the population is subject to multiple factors that affect its size?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is subject to a single factor that affects its size. If the population is subject to multiple factors, you will need to use a more complex formula or model to calculate the remaining population.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to multiple factors that affect its size?
A: If the decrease rate is given as a percentage and the population is subject to multiple factors that affect its size, you will need to use a more complex formula or model to calculate the remaining population. This may involve using a system of equations or a simulation model to account for the multiple factors.
Q: Can I use the formula to calculate the remaining population if the population is subject to random fluctuations?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is subject to a deterministic process. If the population is subject to random fluctuations, you will need to use a more complex formula or model that takes into account the uncertainty in the population size.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to random fluctuations?
A: If the decrease rate is given as a percentage and the population is subject to random fluctuations, you will need to use a more complex formula or model that takes into account the uncertainty in the population size. This may involve using a stochastic model or a simulation model to account for the random fluctuations.
Q: Can I use the formula to calculate the remaining population if the population is subject to external factors that affect its size?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is subject to internal factors that affect its size. If the population is subject to external factors, you will need to use a more complex formula or model that takes into account the external factors.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to external factors that affect its size?
A: If the decrease rate is given as a percentage and the population is subject to external factors that affect its size, you will need to use a more complex formula or model that takes into account the external factors. This may involve using a system of equations or a simulation model to account for the external factors.
Q: Can I use the formula to calculate the remaining population if the population is subject to both internal and external factors that affect its size?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is subject to either internal or external factors, but not both. If the population is subject to both internal and external factors, you will need to use a more complex formula or model that takes into account both types of factors.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to both internal and external factors that affect its size?
A: If the decrease rate is given as a percentage and the population is subject to both internal and external factors that affect its size, you will need to use a more complex formula or model that takes into account both types of factors. This may involve using a system of equations or a simulation model to account for the internal and external factors.
Q: Can I use the formula to calculate the remaining population if the population is subject to non-linear changes in its size?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is subject to linear changes in its size. If the population is subject to non-linear changes, you will need to use a more complex formula or model that takes into account the non-linear changes.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to non-linear changes in its size?
A: If the decrease rate is given as a percentage and the population is subject to non-linear changes in its size, you will need to use a more complex formula or model that takes into account the non-linear changes. This may involve using a system of equations or a simulation model to account for the non-linear changes.
Q: Can I use the formula to calculate the remaining population if the population is subject to multiple populations that interact with each other?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is a single entity that is subject to a single factor that affects its size. If the population is subject to multiple populations that interact with each other, you will need to use a more complex formula or model that takes into account the interactions between the populations.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to multiple populations that interact with each other?
A: If the decrease rate is given as a percentage and the population is subject to multiple populations that interact with each other, you will need to use a more complex formula or model that takes into account the interactions between the populations. This may involve using a system of equations or a simulation model to account for the interactions between the populations.
Q: Can I use the formula to calculate the remaining population if the population is subject to changes in its size that are influenced by external factors?
A: No, the formula P(n) = P(0) x (1 - r)^n assumes that the population is subject to internal factors that affect its size. If the population is subject to changes in its size that are influenced by external factors, you will need to use a more complex formula or model that takes into account the external factors.
Q: How do I calculate the remaining population if the decrease rate is given as a percentage and the population is subject to changes in its size that are influenced by external factors?
A: If the decrease rate is given as a percentage and the population is subject to changes in its size that are influenced by external factors, you will need to use