Answer
The simplified value of the number of tablets and PC sales is $421$ after $5$ years.
Work Step by Step
The worldwide PC and tablet sales function is given by $ f\left( t \right)=33.4{{\left( 1.66 \right)}^{t}}$, where $ t=$ number of years after 2009.
$421$ million PCs were sold according to the model.
Now consider the given details:
$ f\left( t \right)=33.4{{\left( 1.66 \right)}^{t}}$, where $ t $ is the number of years after 2009.
So, $ f\left( t \right)=421\text{ million}$.
Apply this value in the above stated equation of the function as follows:
$\begin{align}
& f\left( t \right)=33.4{{\left( 1.66 \right)}^{t}} \\
& 421=33.4{{\left( 1.66 \right)}^{t}} \\
& {{\left( 1.66 \right)}^{t}}=\frac{421}{33.4} \\
& =12.60
\end{align}$
Thus,
${{\left( 1.66 \right)}^{t}}=12.60$
Apply the logarithm properties on both sides as follows:
$\begin{align}
& t\log \left( 1.66 \right)=\log \left( 12.60 \right) \\
& t=\frac{\log \left( 12.60 \right)}{\log \left( 1.66 \right)} \\
& =4.999 \\
& =5\text{ years}
\end{align}$
Hence, 421 million PCs and tablets were sold after 5 years.
