Question

Compare the doubling times found with the approximate and exact doubling time formulas. Then use the exact doubling time formula to answer the given question. A family of 100 termites invades your house and its population increases at a rate of 21% per week. How many termites will be in your house after 1 year (52 weeks) calculate the doubling times found with the approximate and exact doubling time. The approximate doubling time is ? Weeks and the exact doubling time is ? Weeks round to two decimal places as needed

Answer 1

The approximate doubling time is about 3.33 weeks, while the exact doubling time is slightly lower at 3.21 weeks.

The approximate doubling time formula is:

Doubling Time (approx) = 70 / Growth Rate (%)

In this case, the growth rate is 21%. Plugging it in:

Doubling Time (approx) = 70 / 21% ≈ 3.33 weeks

The exact doubling time formula is:

Doubling Time (exact) = ln(2) / ln(1 + Growth Rate / 100)

Using the natural logarithm (ln) and taking the growth rate as 21%:

Doubling Time (exact) = ln(2) / ln(1 + 21/100) ≈ 3.21 weeks

The approximate doubling time is about 3.33 weeks, while the exact doubling time is slightly lower at 3.21 weeks. This difference is relatively small because the growth rate is not very high (21%). As the growth rate increases, the discrepancy between the two formulas becomes more significant.