Question

a study is done on the population of a certain fish species in a lake. suppose that the population size ( p(t) ) after ( t ) years is given by the following function. ( p(t)=510(1.21)^t ) find the initial population size. does the function represent growth or decay? ( circ ) growth ( circ ) decay by what percent does the population size change each year? ( square% )

Explanation:

Step1: Find initial population (t=0)

Substitute $t=0$ into $P(t)$:
$P(0)=510(1.21)^0 = 510 \times 1 = 510$

Step2: Identify growth/decay

Check base of exponent: $1.21 > 1$, so it is growth.

Step3: Calculate annual percent change

Subtract 1 from the base, convert to percent:
$(1.21 - 1) \times 100\% = 21\%$

Answer:

Initial population size: 510
Growth or decay: growth
Annual percent change: 21%