Question

if the population of a country increases at a rate of 1.5% annually and its current population is 430,000, how many years will it take for the population to triple?
a. 3 years
b. 150 years
c. 2 years
d. 74 years

Explanation:

Step1: Recall the compound - growth formula

The formula for compound growth 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 (in decimal form), and $t$ is the number of years. We want the population to triple, so $P = 3P_0$. The growth rate $r=1.5\%=0.015$. Substituting $P = 3P_0$ into the formula, we get $3P_0=P_0(1 + 0.015)^t$.

Step2: Simplify the equation

Since $P_0
eq0$ (because the initial population is 430,000, which is non - zero), we can divide both sides of the equation $3P_0=P_0(1.015)^t$ by $P_0$. This gives us $3=(1.015)^t$.

Step3: Solve for t using logarithms

To solve for $t$, we take the natural logarithm of both sides of the equation $3=(1.015)^t$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(3)=t\ln(1.015)$. Then, we can solve for $t$ by dividing both sides by $\ln(1.015)$: $t=\frac{\ln(3)}{\ln(1.015)}$.

Step4: Calculate the value of t

We know that $\ln(3)\approx1.0986$ and $\ln(1.015)\approx0.014889$. Then $t = \frac{1.0986}{0.014889}\approx73.79\approx74$ years.

Answer:

D. 74 years