Final answer:
The rate of growth for the population of pavement ants is approximately 13.9%, and the common ratio is approximately 1.149, indicating a 14.9% yearly increase in the population.
Step-by-step explanation:
Finding the Exponential Growth Rate and Common Ratio
To determine the rate of growth and the common ratio for a population of pavement ants in Taylorsville, UT, we will use the formula for exponential growth:
P(t) = P0ert
Where:
P(t) is the population at time t,
P0 is the initial population,
r is the rate of growth, and
t is the time in years.
The initial population (P0) is 3000, the population at time t (P(t)) is 24000, and the time t is 16 years. Plugging in the values, we get:
24000 = 3000e16r
We can solve for r:
e16r = 24000 / 3000 = 8
16r = ln(8)
r = ln(8) / 16
After calculating, r is approximately 0.1386. To convert this to a percentage, we multiply by 100, getting 13.9%.
The common ratio is the base of the exponent e raised to the power of r. Since er is how much the population multiplies by each year, we've found it to be e0.1386, which calculates to approximately 1.149, or a 14.9% increase each year.