Question

Use the approximate​ half-life formula for the case described below. Discuss whether the formula is valid for the case described. Poaching is causing a population of elephants to decline by 4 ​% per year. What is the​ half-life for the​ population? If there are​ 10,000 elephants​ today, how many will remain in 50 ​years? Question content area bottom Part 1 Use the approximate​ half-life formula. What is the​ half-life for the​ population? enter your response here years ​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)

Answer 1

The approximate half-life formula is:

t(1/2) ≈ 0.693 / r

where r is the annual rate of decline as a decimal. In this case, r = 0.04, since the population is declining by 4% per year. Therefore:

t(1/2) ≈ 0.693 / 0.04 ≈ 17.325 years

So the half-life for the population of elephants is approximately 17.325 years.

The approximate half-life formula is valid for this case since the rate of decline is relatively small (4% per year) and the formula is designed for exponential decay with small rates of change. However, it is important to note that the formula is only an approximation and may not be accurate for large rates of change or over long periods of time.

To find the number of elephants that will remain in 50 years, we can use the formula:

N(t) = N0 * (1 - r)^t

where N0 is the initial population (10,000), r is the annual rate of decline as a decimal (0.04), and t is the time in years (50). Therefore:

N(50) = 10,000 * (1 - 0.04)^50 ≈ 4,877

So there will be approximately 4,877 elephants remaining in 50 years. Answer: 4,877.