Question

type the correct answer in the box. use numerals instead of words.
a population of beetles increases by 5% every year. if at the start of the year the population is at 10,000 beetles, what will its population be after three years?
at the end of the third year the population of beetles will be

Explanation:

Step1: Recall the compound growth formula

The formula for compound growth is \( P = P_0(1 + r)^n \), where \( P_0 \) is the initial population, \( r \) is the growth rate (as a decimal), and \( n \) is the number of years. Here, \( P_0 = 10000 \), \( r = 0.05 \) (since 5% = 0.05), and \( n = 3 \).

Step2: Substitute the values into the formula

Substitute \( P_0 = 10000 \), \( r = 0.05 \), and \( n = 3 \) into the formula: \( P = 10000(1 + 0.05)^3 \).
First, calculate \( 1 + 0.05 = 1.05 \). Then, calculate \( 1.05^3 = 1.05 \times 1.05 \times 1.05 = 1.157625 \).
Finally, multiply by the initial population: \( P = 10000 \times 1.157625 = 11576.25 \). Since we are dealing with a population, we can consider the whole number, so it's 11576 (or 11576.25 if decimal is allowed, but typically populations are whole numbers, though the problem doesn't specify, so we'll use the exact value from the calculation).

Answer:

11576.25 (or 11576 if we take the whole number, but the calculation gives 11576.25)