Answer
a) The percentage model function for African Americans who used cigarettes is $ y=38-0.54x $.
b) In the year $1993$, the use of cigarettes for both groups was the same. The percentage of use of cigarettes for African Americans in 1993 is, $\text{33}\text{.68 }\!\!%\!\!\text{ }$. The percentage of use of cigarettes for Hispanics in 1993 is, $\text{33}\text{.68 }\!\!%\!\!\text{ }$.
Work Step by Step
(a)
We use the model function of cigarette use by African Americans after 1985:
$ y=38-0.54x $
(b)
Step 1: Consider the percentage model function for African Americans who used cigarettes after x years from $1985$:
$ y=38-0.54x $ (I)
Step 2: The percentage model function for Hispanics who used cigarettes after x years from $1985$ is
$0.79x+y-40$
Rewrite this function:
$\begin{align}
& 0.79x+y-40=0 \\
& y=40-0.79x
\end{align}$ (II)
Step 3: After x years from $1985$, the use of cigarettes would be the same. Therefore, equate both functions and find the value of x.
$\begin{align}
& 38-0.54x=40-0.79x \\
& 0.79x-0.54x=40-38 \\
& 0.25x=2 \\
& x=8
\end{align}$
Since, $\left( x=8 \right)$ then after $8$ (1993) years the use of cigarettes was the same for Hispanics and African Americans.
Step 4: Compute the percentage of the use of cigarettes for African Americans in 1993 by substituting $\left( x=8 \right)$ in equation (I) as shown below:
$\begin{align}
& y=38-0.54\times 8 \\
& =38-4.32 \\
& =33.68
\end{align}$
Since, $y=33.68$.
Therefore, the percentage of cigarette use by African Americans in the year $1993$ is $\text{33}\text{.68 }\!\!%\!\!\text{ }$.
Step 5: Find the percentage of use of cigarettes for Hispanics, also in $1993$, by substituting $\left( x=8 \right)$ in equation (II) as shown below:
$\begin{align}
& y=40-0.79\times 8 \\
& =40-6.32 \\
& =33.68
\end{align}$
Since, $ y=33.68$
Therefore, the percentage of use of cigarettes for Hispanics in the year $1993$ is $\text{33}\text{.68 }\!\!%\!\!\text{ }$.
Thus, the use of cigarettes for both groups was the same after $8$ years from the year $1985$.
