A species of beetles grows 32% every year. Suppose 100 beetles are released into a field.
How many beetles will there be in 10 years?
what is
100(1.32)^10 ??
To calculate the number of beetles after 10 years, we need to apply the growth rate of 32% to the initial population of 100 beetles for each of the ten years.
Step 1: Calculate the growth rate
Growth rate = 32%
Step 2: Calculate the number of beetles after each year
Year 1: 100 + (32% of 100) = 100 + 32 = 132 beetles
Year 2: 132 + (32% of 132) = 132 + 42.24 = 174.24 ≈ 174 beetles
Year 3: 174 + (32% of 174) = 174 + 55.68 = 229.68 ≈ 230 beetles
Year 4: 230 + (32% of 230) = 230 + 73.6 = 303.6 ≈ 304 beetles
Year 5: 304 + (32% of 304) = 304 + 97.28 = 401.28 ≈ 401 beetles
Year 6: 401 + (32% of 401) = 401 + 128.32 = 529.32 ≈ 529 beetles
Year 7: 529 + (32% of 529) = 529 + 169.28 = 698.28 ≈ 698 beetles
Year 8: 698 + (32% of 698) = 698 + 223.36 = 921.36 ≈ 921 beetles
Year 9: 921 + (32% of 921) = 921 + 294.72 = 1215.72 ≈ 1216 beetles
Year 10: 1216 + (32% of 1216) = 1216 + 389.12 = 1605.12 ≈ 1605 beetles
Therefore, after 10 years, there will be approximately 1605 beetles in the field.
To find the number of beetles in 10 years, we can use the formula for compound interest, which is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/quantity
P = the initial quantity
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, the initial quantity of beetles is 100, the annual growth rate is 32% (or 0.32 as a decimal), and the number of years is 10. Since the beetles are growing continuously, we can assume that interest is compounded infinitely, which means n approaches infinity.
Plugging in the values into the formula, we get:
A = 100(1 + 0.32/1)^(1*10)
Simplifying the equation, we have:
A = 100(1 + 0.32)^10
Calculating the expression inside the parentheses first:
(1 + 0.32) = 1.32
Now, substituting back into the formula:
A = 100(1.32)^10
Using a calculator, we find:
A ≈ 932.79
Therefore, there will be approximately 932.79 beetles in the field after 10 years.