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
- 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. - 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
- 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. - 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
- 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. - 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
- 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. - 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
- 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. - 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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