Answer
4 seconds
Work Step by Step
1. The clue here is to figure out that telescope is going to have the same rate of rotation as the earth that is simply $360^{\circ}/24hours$
2. Don't get confused by minutes of the time and the minutes of a degree. We are going to convert degrees to minutes (')
$\frac{360^{\circ}}{24hr}\times\frac{69'}{1^{\circ}}=21600'/24hr$
3. Let's reduce the number
$\frac{21600'}{24h}=\frac{900'}{1hr}$
4. We need to get down to one minute of a degree, so let's start applying conversation factors and reducing
$\frac{900'}{1hr}\times \frac{1hr}{60min}$= 15'/min
5. Finally, how much does it take to go through one minute of rotation
$\frac{15'}{60sec} =\frac{1'}{4sec}$
