# The world's population was 5.51 billion on January 1, 1993 and 5.88 billion on January 1, 1998....

## Question:

The world's population was 5.51 billion on January 1, 1993 and 5.88 billion on January 1, 1998. Assume that at any time the population grows at a rate proportional to the population at that time. In what year should the world's population reach 7 billion?

## Exponential Growth:

If at a general time {eq}t {/eq}, a quantity {eq}P {/eq} grows at a rate {eq}\dfrac{\mathrm{d}P}{\mathrm{d} t} {/eq} proportional to the amount of the quantity {eq}P {/eq} at {eq}t=t {/eq}, then denoting the constant of this proportionality as {eq}r {/eq}, it holds true that {eq}\dfrac{\mathrm{d}P}{\mathrm{d} t} = rP {/eq}.

If {eq}P=P_0 {/eq} at {eq}t=0 {/eq}, then {eq}P(t)=P_0e^{rt} {/eq}.

Given:

The world's population in 1993, that is, at {eq}t=0\mathrm{\,yr} {/eq} is {eq}P_0=5.51 \mathrm{\,BB} {/eq}.

{eq}\dfrac{\mathrm{d}P}{\mathrm{d} t} {/eq} is proportional to the population {eq}P {/eq}. If {eq}r {/eq} is the constant of proportionality, then

{eq}\begin{align*} P(t)&=P_0e^{rt}\\[2ex] \Rightarrow P(t)&=5.51e^{rt}\\[2ex] \Rightarrow P(5)&=5.51e^{r*5}&\left[\text{Setting }t=5\mathrm{\,yr} \text{ for the population in 1998}\right]\\[2ex] \Rightarrow 5.88&= 5.51e^{5r}\\[2ex] \Rightarrow e^{5r} &= \frac {588}{551}\\[2ex] \Rightarrow \ln e^{5r} &= \ln \frac {588}{551}\\[2ex] \Rightarrow 5r &= \ln \frac {588}{551}\\[2ex] \Rightarrow r &= \left(\frac 15\right) \ln \frac {588}{551} \end{align*} {/eq}

So when {eq}P(t)=7 \mathrm{\,BB} {/eq},

{eq}\begin{align*} P(t)&=5.51e^{rt}\\[2ex] 7&=5.51e^{rt}\\[2ex] \Rightarrow e^{rt}&=\frac {7}{5.51}\\[2ex] \Rightarrow rt&=\ln \frac {700}{551}\\[2ex] \Rightarrow t&=\left(\frac 1r\right)\ln \frac {700}{551}\\[2ex] \Rightarrow t&=\left(\frac 1{\left(\frac 15\right) \ln \frac {588}{551}}\right)\ln \frac {700}{551}\\[2ex] \Rightarrow t&\approx 18.41\mathrm{\,yr} \end{align*} {/eq}

{eq}18.41\mathrm{\,yr} {/eq} after Jan 1, 1993, we should be in {eq}1993+18=\color{Blue}{2011\hspace{0.8cm}\mathrm{(Answer)}} {/eq}.