Question

type the correct answer in the box. use numerals instead of words.
mitten crabs are an invasive species in north america. a coastal ecosystem has a population of 320 mitten crabs. the population is predicted to increase by 50% per year.
the population will reach an estimated 1,620 crabs in 　 years.

Explanation:

Step1: Define growth formula

This is exponential growth, so use $P(t) = P_0(1+r)^t$, where $P_0=320$, $r=0.5$, $P(t)=1620$.

Step2: Substitute values into formula

$$1620 = 320(1+0.5)^t$$

Step3: Simplify the equation

Divide both sides by 320: $\frac{1620}{320} = 1.5^t$, so $\frac{81}{16} = 1.5^t$

Step4: Rewrite terms as powers of 1.5

Note $1.5=\frac{3}{2}$, $\frac{81}{16}=(\frac{3}{2})^4$, so $(\frac{3}{2})^4 = (\frac{3}{2})^t$

Step5: Solve for t

Since bases are equal, exponents match: $t=4$

Answer:

4