At noon, the minute and hour hands of a clock overlap. In how many minutes will they overlap again?
The minute hand of a clock completes a full revolution every 60 minutes, while the hour hand completes a full revolution every 12 hours, or 720 minutes.
Since the minute hand moves 12 times faster than the hour hand, it will take $\frac{720}{60} = 12$ rotations for the minute and hour hands to overlap again. Therefore, the hands will overlap in $\frac{12}{12} = \boxed{1}$ minute.
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I apologize for the confusion. Let's try again.
In 60 minutes, the minute hand moves 360 degrees, while the hour hand moves $\frac{1}{2}$ of that, or 180 degrees.
Let's say the minute hand is at 12 and the hour hand is at some position $x$ between the 11 and 12 o'clock positions. In 60 minutes, the minute hand moves 360 degrees, so it moves 30 degrees for every 5-minute interval. The hour hand also moves 30 degrees in 60 minutes, so it moves 0.5 degrees every minute.
We want to find how many minutes it takes for the hour hand to intersect with the minute hand, or how much the hour hand needs to travel to equal the distance between 12 and $x$, which is $360 - x$ degrees.
Since the hour hand moves 0.5 degrees every minute, we have the equation $0.5m = 360 - x$, where $m$ is the number of minutes.
Solving for $m$, we have:
$0.5m = 360 - x$
$m = 720 - 2x$
Since $x$ can be any number between 0 and 30 (since the hour hand moves halfway between the hour marks in a 30-minute interval), $m$ can take on any value between 720 and 0. Therefore, the hour and minute hands overlap every $\boxed{720}$ minutes.
At noon, the minute and hour hands of a clock overlap. They will overlap again in 1 hour and x minutes. Find x.
When the hands of a clock overlap, it means that they are pointing to the same position on the clock.
At noon, the hour hand points to the 12 and the minute hand is on the 12 as well (since they overlap).
To determine when they will overlap again, we need to find the time it takes for the minute hand to travel a full revolution of 360 degrees while the hour hand moves to the same position.
The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute.
The hour hand moves 30 degrees in 60 minutes, so it moves 0.5 degrees per minute.
Since the minute hand moves 6 degrees per minute and the hour hand moves 0.5 degrees per minute, the minute hand gains 5.5 degrees on the hour hand every minute.
Since they overlap at noon and the minute hand needs to gain 360 degrees on the hour hand, it will take $\frac{360}{5.5} \approx 65.45$ minutes.
Thus, they will overlap again in approximately 1 hour and 5.45 minutes.
Therefore, x is approximately $\boxed{5}$ minutes.
One of the lines formed by two intersecting line measures 42˚. What are the measures of the other three angles?
When two lines intersect, they form four angles. If one of these angles measures 42 degrees, the other three angles can be calculated.
Since the angles around a point add up to 360 degrees, we subtract the given angle from 360 to find the sum of the other three angles: $360 - 42 = 318$ degrees.
The other three angles can be any combination that adds up to 318 degrees. They can be equal or different.
Some possible combinations for the measures of the other three angles include:
Angle 2: 90 degrees
Angle 3: 110 degrees
Angle 4: 118 degrees
Angle 2: 60 degrees
Angle 3: 140 degrees
Angle 4: 118 degrees
Angle 2: 80 degrees
Angle 3: 120 degrees
Angle 4: 118 degrees
These are just a few examples, but there are many different combinations of angles that add up to 318 degrees.