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5. Uranium-238, the most common uranium isotope, has a half-life of 4.5 billion years. After 10,000 years, what percentage of the original amount of uranium remains?

6. Mouse populations grow exponentially when left unchecked. Assuming that 6 mice are left alone, how many mice will exist in 3 months if the mice can increase their population by 200% every 3 weeks?

Answer :

To solve the problem of determining how many mice will exist in 3 months if the population increases by 200% every 3 weeks, let's break it down step by step:

1. Initial Population:
- Start with 6 mice.

2. Growth Rate:
- An increase of 200% means the population triples (100% initial population + 200% increase = 300% of original, or 3 times the original number).

3. Time Frame:
- We need to calculate how many complete 3-week periods fit into 3 months.
- There are approximately 4 weeks in a month, so 3 months have about 12 weeks.
- In 12 weeks, there are [tex]\(12 \div 3 = 4\)[/tex] complete 3-week periods.

4. Exponential Growth:
- Each 3-week period, the population triples.
- So, after 4 periods, the population will be [tex]\(6 \times 3^4\)[/tex].

5. Calculate the Final Population:
- [tex]\(3^4 = 81\)[/tex]
- Therefore, the number of mice is [tex]\(6 \times 81 = 486\)[/tex].

So, after 3 months, there will be 486 mice.

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