Answer
The required solution is $\text{1047}\ \text{mph}$
Work Step by Step
The angular speed $\omega $ is $\frac{\pi }{12}\ \text{radians per hour}$.
The radius of the earth $ r $ is $4000\ \text{miles}$.
The linear speed $ v $ is given by
$ v=r\omega $
Put $4000\ \text{miles}$ for $ r $ and $\frac{\pi }{12}\ \text{radians per hour}$ for $\omega $:
$\begin{align}
& v=\left( 4000\ \text{miles} \right)\left( \frac{\pi }{12}\ \text{radians per hour} \right) \\
& =\frac{1000\pi }{3}\ \text{miles}\ \text{per hour}
\end{align}$
Put $\pi =3.14159$:
$\begin{align}
& v=\frac{1000\left( 3.14159 \right)}{3}\ \text{miles}\ \text{per hour} \\
& \text{=1047}\text{.19}\ \text{miles}\ \text{per hour}\approx \text{1047}\ \text{miles}\ \text{per hour}
\end{align}$
