Question

population the population of species a is decreasing at a rate of about 0.25% per quarter. the population of species b is decreasing at a rate of about 1.34% per year. determine which species has a population that is decreasing at a faster rate. explain. select choice is decreasing at a faster rate. species a is decreasing at a rate of 0.25 per quarter. species b is decreasing at a rate of 0.34 per quarter.

Explanation:

Step1: Convert annual rate to quarterly rate

To compare the rates, we convert Species B's annual 1.34% decrease to a quarterly rate. Use the effective quarterly rate formula for exponential decay:
Let \( r_{annual} = 0.0134 \), \( n = 4 \) (quarters per year).
The quarterly rate \( r_{quarterly} \) satisfies \( (1 - r_{quarterly})^4 = 1 - r_{annual} \).
Rearrange to solve for \( r_{quarterly} \):
$$r_{quarterly} = 1 - (1 - 0.0134)^{\frac{1}{4}}$$
Calculate \( (0.9866)^{\frac{1}{4}} \approx 0.99665 \), so \( r_{quarterly} \approx 1 - 0.99665 = 0.00335 \), or ~0.34% per quarter.

Step2: Compare quarterly rates

Species A's quarterly decrease rate is 0.25%, Species B's is ~0.34%. Since \( 0.34\% > 0.25\% \), Species B's rate is faster.

Converting Species B's 1.34% annual decrease to a quarterly rate gives approximately 0.34% per quarter, which is higher than Species A's 0.25% quarterly decrease rate.

Answer:

Species B is decreasing at a faster rate.
Species A is decreasing at a rate of 0.25 per quarter.
Species B is decreasing at a rate of 0.34 per quarter.