To determine the angle between the minute and hour hands at 8:30, we perform the following calculations:- At 8:00, the hour hand points to the 8, and the minute hand points to the 12.- The angle subtended by each hour mark on the clock face is \( \frac{360^\circ}{12} = 30^\circ \).- Consequently, at 8:00, the hour hand's position is \( 8 \times 30^\circ = 240^\circ \) relative to the 12 o'clock position.- At 8:30, the minute hand is at the 6 (corresponding to 30 minutes), which is \( 180^\circ \) from the 12 o'clock position.- The angle between the hour and minute hands is calculated as the absolute difference between their positions: \( |240^\circ - 180^\circ| = 60^\circ \).Therefore, the angle between the two hands is \( 240^\circ - (180^\circ + 30^\circ) = 30^\circ \). This calculation is incorrect. The correct calculation is: The hour hand moves \( 0.5^\circ \) per minute. So at 8:30, the hour hand is at \( 240^\circ + (30 \times 0.5^\circ) = 240^\circ + 15^\circ = 255^\circ \). The minute hand is at \( 180^\circ \). The angle between them is \( |255^\circ - 180^\circ| = 75^\circ \). This is also incorrect. Let's re-evaluate. The hour hand moves 360 degrees in 12 hours, so it moves 30 degrees per hour. In 30 minutes, it moves half of that, which is 15 degrees. So at 8:30, the hour hand is at \( 8 \times 30^\circ + 15^\circ = 240^\circ + 15^\circ = 255^\circ \) from the 12 o'clock position.The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute. At 30 minutes past the hour, the minute hand is at \( 30 \times 6^\circ = 180^\circ \) from the 12 o'clock position.The difference between their positions is \( |255^\circ - 180^\circ| = 75^\circ \).This is still not correct. Let's use the formula directly: Angle = \( |30H - \frac{11}{2}M| \), where H is the hour and M is the minute.For 8:30, H = 8 and M = 30.Angle = \( |30(8) - \frac{11}{2}(30)| \)Angle = \( |240 - 11(15)| \)Angle = \( |240 - 165| \)Angle = \( 75^\circ \).There appears to be an error in the provided original text's calculation. The angle between the hands at 8:30 is actually 75 degrees. The provided answer of 105 degrees is incorrect based on standard clock angle calculations. The original text concludes with \( 180^\circ - 60^\circ = 105^\circ \), which seems to be a miscalculation. The difference was calculated as \( 240^\circ - 180^\circ = 60^\circ \), and then \( 180^\circ - 60^\circ \) is used. This is not a valid step. The correct difference between the positions is \( 60^\circ \). However, this is not the final angle. The final angle calculation appears flawed. Let's assume the initial calculation of \( 240^\circ \) and \( 180^\circ \) is correct for their base positions. The issue is how the final angle is derived.Revisiting the problem, the hour hand is at \( 240^\circ \) at 8:00. The minute hand is at \( 0^\circ \) (or \( 360^\circ \)).At 8:30:Hour hand moves \( 30^\circ/hour * 0.5 hour = 15^\circ \) past the 8.So, hour hand position = \( 240^\circ + 15^\circ = 255^\circ \).Minute hand position = \( 30 minutes * 6^\circ/minute = 180^\circ \).Angle = \( |255^\circ - 180^\circ| = 75^\circ \). The original text's calculation resulting in 105° is erroneous.