Answer
a) $N=84+9.818 t$
The slope of the function indicates the number of people suffering from asthma has increased by 9.818 million people per.
b) $N=84(1.0596)^t$
c) Linear: $378.54$ million asthma sufferers
Exponential: $477.021$ million asthma sufferers
Work Step by Step
A)Le want $N(t)=b+mt$. The slope of the line is
$$
m=\frac{\Delta N}{\Delta t}=\frac{300-84}{2012-1990}=\frac{216}{22}=9.818
$$
We know that $N=84$ when $t=0$, the vertical intercept is 84 and the linear formula is
$$
N=84+9.818 t
$$
The slope of the function indicates the number of people suffering from asthma has increased by 9.818 million people per year.
B) We know that $N=84$ when $t=0$. This gives the exponential function $N=84 b^t$ . Since $N=300$ when $t=22$ we have
$$
\begin{aligned}
N & =84 b^t \\
300 & =84 b^{22} \\
b^{22} & =\frac{300}{84}=3.5714 \\
b & =(3.5714)^{1 / 22}=1.0596
\end{aligned}
$$
The required function is
$$
N=84(1.0596)^t
$$
C) For the year 2020, we set $t=30$ into the both functions to predicted the number of asthma sufferers in 2020. For the linear function, we have $N=84+9.818(30)=378.54$ million asthma sufferers, and $N=84(1.0596)^{30}=477.021$ million asthma sufferers.
