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------------------------------------------------ 1. Suppose that the population of a certain country grows at an annual rate of 4%. If the current population is 3 million:

a) What will the population be in 10 years?

b) How long will it take the population to reach 8 million?

2. If the current population is 10 million, and it grows to 25 million after 15 years, calculate the growth rate for this population.

3. The doubling time of a population of flies is eight days. If there are initially 100 flies:

a) How many flies will there be in 17 days?

b) How many flies will there be in two months?

4. Suppose that a bacteria population doubles every six hours. If the initial population is 4000 individuals:

a) Calculate the number of bacteria after two days.

b) How many hours would it take the population to increase to 25,000?

5. The doubling time of a population of bacteria is 6 minutes. If there are initially 40 bacteria:

a) How many bacteria will there be in one hour?

b) How long would it take the population to increase to 1280?

6. If there are currently 5000 elephants left in the world, and it has a half-life of 20 years:

a) How many will remain in 60 years?

b) How long would it take the population to decrease to 156?

7. Plutonium has a half-life of 24,000 years. Suppose that 30 pounds of it was dumped at a nuclear waste site. How long would it take for it to decay into 2 pounds?

Answer :

Final answer:

In these population growth and decay problems, we use exponential growth and decay formulas to calculate the future population or time it takes for a population to reach a certain number. For example, for the given population growth rate and current population, we can find the population in a certain number of years using the formula P = P0 * (1 + r)^t. For population doubling time problems, we use the formula P = P0 * (2^(t/d)), where t is the time and d is the doubling time.


Explanation:

1. Population Growth

  1. In 10 years, the population will increase by 4% annually.
    Population after 10 years = 3 million * (1 + 0.04)^10 = 4.37 million.
    Similarly, to find the time taken for the population to reach 8 million:
    8 million = 3 million * (1 + 0.04)^n
    Solving for n, we get n = 21.62 years. So, it will take approximately 22 years for the population to reach 8 million.

  3. To calculate the growth rate, we use the formula:
    Growth rate = (Final population - Initial population) / (Initial population) * 100
    Growth rate = (25 million - 10 million) / (10 million) * 100
    Growth rate = 150%.

2. Population Doubling Time

  1. In 17 days, the population doubles every 8 days.
    Number of doubles = 17 days / 8 days = 2.125.
    Number of flies after 17 days = Initial population * (2^Number of doubles) = 100 * (2^2.125) = 313.84 flies. Approximately 314 flies.

  3. In 2 months, there are 30 days.
    Number of doubles = 30 days / 8 days = 3.75.
    Number of flies after 2 months = Initial population * (2^Number of doubles) = 100 * (2^3.75) = 751.31 flies. Approximately 751 flies.

3. Bacteria Population

  1. After 2 days, the population doubles every 6 hours.
    Number of doubles = 2 days * 24 hours / 6 hours = 8.
    Number of bacteria after 2 days = Initial population * (2^Number of doubles) = 4000 * (2^8) = 1,024,000 bacteria.

  3. To calculate the number of hours to reach 25,000 population:
    Number of doubles = Number of hours / 6 hours = 25,000 / 4000 = 6.25.
    Number of hours = Number of doubles * 6 = 6.25 * 6 = 37.5 hours. Approximately 38 hours.

4. Bacteria Population

  1. In one hour, the population doubles every 6 minutes.
    Number of doubles = 1 hour * 60 minutes / 6 minutes = 10.
    Number of bacteria in one hour = Initial population * (2^Number of doubles) = 40 * (2^10) = 40 * 1024 = 40,960 bacteria.

  3. To calculate the time to reach 1280 population:
    Number of doubles = Time in minutes / 6 minutes = 1280 / 40 = 32.
    Number of hours = Number of doubles * 6 = 32 * 6 = 192 minutes. Approximately 3 hours and 12 minutes.

5. Elephant Population

  1. After 60 years, the population halves every 20 years.
    Number of halves = 60 years / 20 years = 3.
    Number of elephants remaining after 60 years = Initial population * (0.5^Number of halves) = 5000 * (0.5^3) = 5000 * 0.125 = 625 elephants.

  3. To calculate the time to reach 156 population:
    Number of halves = Time in years / 20 years = 5000 / 156 = 32.051.
    Time in years = Number of halves * 20 = 32.051 * 20 = 641.02 years. Approximately 641 years.

6. Plutonium Decay

Plutonium has a half-life of 24,000 years. To calculate the time it takes for 30 pounds of plutonium to decay into 2 pounds:
Number of half-lives = log(2 pounds / 30 pounds) / log(0.5) = 4.75.
Time in years = Number of half-lives * 24,000 = 4.75 * 24,000 = 114,000 years.


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