Question

initially, a population is 3000, and it grows by 3% each year. the population is an exponential function of time because the percentage change of population is not constant. it has an initial population size. the percentage change of population is the same each year. it does not represent rapid growth. find a formula for the population n at any time. (express numbers in exact form. use symbolic notation and fractions where needed. let t be the time in years.) n = what is the population size after five years? (round your result to the nearest whole number.) population size:

Explanation:

Step1: Recall Exponential Growth Formula

The general formula for exponential growth is \( N(t) = N_0(1 + r)^t \), where \( N_0 \) is the initial population, \( r \) is the growth rate (as a decimal), and \( t \) is time in years. Here, \( N_0 = 3000 \) and \( r = 0.03 \) (since 3% = 0.03).

Step2: Formulate the Population Function

Substitute \( N_0 = 3000 \) and \( r = 0.03 \) into the formula: \( N(t) = 3000(1 + 0.03)^t = 3000(1.03)^t \).

Step3: Calculate Population After 5 Years

Substitute \( t = 5 \) into the formula: \( N(5) = 3000(1.03)^5 \). First, calculate \( (1.03)^5 \approx 1.159274 \). Then, multiply by 3000: \( 3000 \times 1.159274 \approx 3477.822 \). Round to the nearest whole number.

Answer:

For the formula: \( N = 3000(1.03)^t \)
For the population after five years: \( 3478 \)