Question

question 9 (mandatory) (1 point)
the population of a small village has grown at an annual rate of approximately 5.5%. how long will it take for its population of 3200 people to double at this growth rate?

a) about 16 years
b) about 1.5 years
c) about 11 years
d) about 13 years

Explanation:

Step1: Use compound - growth formula

The compound - growth formula for population is $P = P_0(1 + r)^t$, where $P$ is the final population, $P_0$ is the initial population, $r$ is the annual growth rate as a decimal, and $t$ is the number of years. We want to find $t$ when $P = 2P_0$ and $r=0.055$. Substituting into the formula gives $2P_0=P_0(1 + 0.055)^t$.

Step2: Simplify the equation

Divide both sides of the equation $2P_0=P_0(1 + 0.055)^t$ by $P_0$ (since $P_0
eq0$), we get $2=(1.055)^t$.

Step3: Take the natural logarithm of both sides

$\ln(2)=\ln(1.055^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(2)=t\ln(1.055)$.

Step4: Solve for $t$

$t=\frac{\ln(2)}{\ln(1.055)}$. Calculate $\ln(2)\approx0.6931$ and $\ln(1.055)\approx0.0535$. Then $t=\frac{0.6931}{0.0535}\approx12.955\approx13$ years.

Answer:

D. about 13 years