The difference in the length of a mean solar day and a sidereal day is about

The correct option is 4 minutes.

To understand the difference in length between a mean solar day and a sidereal day, we must look at how Earth rotates relative to different reference points:
1. A sidereal day is the time it takes for the Earth to complete one full rotation (360 degrees) on its axis relative to the distant "fixed" stars. This takes approximately 23 hours, 56 minutes, and 4 seconds (about 86,164 seconds).
2. A mean solar day is the time it takes for the Earth to rotate so that the Sun appears at the same position in the sky (e.g., meridian transit) from one day to the next. This is exactly 24 hours (86,400 seconds).

The difference arises because, while the Earth is rotating on its axis, it is also orbiting the Sun. In the course of one day, the Earth travels approximately 1 degree along its orbit around the Sun (since a full orbit is 360 degrees and takes about 365.25 days):

$\frac{360°}{365.25\text{days}}\approx 0.986°\text{per day}$

Consequently, after completing one 360-degree rotation (a sidereal day), the Earth must rotate an extra angle of approximately 0.986 degrees for the Sun to return to the same position in the sky.
Since the Earth rotates 360 degrees in 24 hours, the time required to rotate this extra 0.986 degrees is computed as:

$\text{Extra Time}=24\text{hours}\times \frac{0.986°}{360°}\approx 0.0657\text{hours}$

Converting this value into minutes:

$0.0657\text{hours}\times 60\text{minutes/hour}\approx 3.94\text{minutes}$

Rounding this to the nearest whole minute gives 4 minutes.