Answer
a)
The average rate of population growth in 1920 $=16$ million/ year.
The average rate of population growth in 1980 $=78.5$ million/ year.
b)
$a\approx-0.0002849003$
$b\approx0.52243312243$
$c\approx-6.395641396$
$d\approx1720.586081$
c)
$p'(t)=3at+2bt+c$ (in million population per year)
d)
$p'(20)\approx14.16$ million/year
$p'(80)\approx71.72$ million/year
Hence, these estimates are smaller than in part (a)
e)
$f'(t)=pq^tlnq$
f)
$f'(20)\approx26.25$ (much greater than the estimate value of (a) and (d))
$f'(80)\approx60.28$ (much smallar than the estimate value of (a) and (d))
g)
$p'(85)\approx76.24$ million/ year.
and,
$f'(85)\approx 64.61$ million/ year.
hence, the first estimate is probably accurate.
Work Step by Step
a) The rate of population growth in 1920 and in
1980 :
In 1920:
By using slope method:
$m_1=\frac{1860-1750}{1920-1910}$
$m_1=\frac{110}{10}$
$m_1=11$
similarly,
$m_2=\frac{2070-1860}{1930-1920}$
$m_2=\frac{210}{10}$
$m_2=21$
Averaging the slope :
$\frac{m_1+m_2}{2}=\frac{11+21}{2}$
$\frac{m_1+m_2}{2}=\frac{32}{2}$
$\frac{m_1+m_2}{2}=16 $ million/ year
The average rate of population growth in 1920 $=16$ million/ year.
In 1980:
$m_1=\frac{4450-3710}{1980-1970}$
$m_1=\frac{740}{10}$
$m_1=74$
similarly,
$m_2=\frac{5280-4450}{1990-1980}$
$m_2=\frac{830}{10}$
$m_2=83$
Averaging the slope :
$\frac{m_1+m_2}{2}=\frac{74+83}{2}$
$\frac{m_1+m_2}{2}=\frac{157}{2}$
$\frac{m_1+m_2}{2}=78.5 $ million/ year
The average rate of population growth in 1980 $=78.5$ million/ year.
b) The polynomial function is : $p(t)=at^3+bt^2+ct+d$ (in million
population)
where, $a\approx-0.0002849003$
$b\approx0.52243312243$
$c\approx-6.395641396$
$d\approx1720.586081$
C)
The polynomial function is : $p(t)=at^3+bt^2+ct+d$
Taking differentiate w.r.t $t$.
$p'(t)=\frac{d}{dt}(at^3+bt^2+ct+d)$
$p'(t)=3at+2bt+c+0$
$p'(t)=3at+2bt+c$ (in million population per year)
d)
Estimate the rates of growth in 1920 and
1980:
The 1920 corresponds to $t=20$
$p'(20)\approx14.16$ million/year
similarly, the 1980 corresponds to $t=80$
$p'(80)\approx71.72$ million/year
Hence, these estimates are smaller than in part (a)
e)
Modeling $p(t)$ with the exponential
function
by using $f(t)=pq^t$
where, $p=1.23653{\times}10^9$ and $q=1.01395$
Differentiate w.r.t $t$
$f'(t)=pq^tlnq$
f)
$f'(20)\approx26.25$ (much greater than the estimate value of (a) and (d))
$f'(80)\approx60.28$ (much smallar than the estimate value of (a) and (d))
g)
Estimate the rate of growth in 1985:
$p'(85)\approx76.24$ million/ year.
and,
$f'(85)\approx 64.61$ million/ year.
hence, the first estimate is probably accurate.
