Final answer:
The minute and hour hands of a clock starting at noon will first overlap shortly after 1:05 p.m. The hands overlap when the minute hand gains 360° on the hour hand, which occurs every 65 minutes. The exact time can be calculated using the angles moved by each hand per minute.
Step-by-step explanation:
The question about when the minute and hour hands of a clock overlap after starting at noon is a common problem in mathematics, specifically in the area related to time and angles in clocks. The minute hand moves 360° in 60 minutes, whereas the hour hand moves 360° in 12 hours. This means that for every minute that passes, the minute hand moves 6° (360°/60) and the hour hand moves 0.5° (360°/720). Since they start together at noon, they will overlap when the minute hand has gained a full circle (360°) on the hour hand.
Generally, the hands overlap every 65 minutes. The first time they overlap after noon is shortly after 1:05 p.m. If we wanted to calculate it exactly, we'd set up an equation where the angle of the minute hand is 6 times the number of minutes, and the angle of the hour hand is 0.5 degrees times 60 plus the number of minutes. By setting these two angles equal, we would solve for the time when they overlap.
In practice, though, this is commonly approximated and observed around 1:05 p.m. for the first overlap after noon. If we were to consider this problem in a more precise manner, we would account for the slight variations in the actual solar day compared to our standard clock time, as discussed in the context of sundials and standard time.