Question

question 18 (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 1.5 years
b) about 11 years
c) about 16 years
d) about 13 years

Explanation:

Step1: Use the compound - growth formula

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

Step2: Simplify the equation

Divide both sides of the equation $2P=P(1 + 0.055)^t$ by $P$ (since $P
eq0$). 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 $t$

$t=\frac{\ln(2)}{\ln(1.055)}$. Since $\ln(2)\approx0.693$ and $\ln(1.055)\approx0.0535$, then $t=\frac{0.693}{0.0535}\approx12.95\approx13$ years.

Answer:

d) about 13 years