To the nearest second, what is the first time after 12: 00 that the hour hand and the minute han

step 1 Hello, everyone. This is number 17 and is your geometry textbook. This problem deals with angle

To the nearest second, what is the first time after 12: 00...

Key Concepts

Relative Motion

This concept involves analyzing the movement of one object in relation to another. In the context of clock problems, it is used to determine how quickly one hand catches up to or moves away from the other by considering the difference in their angular velocities.

Clock Angle Problems

This is a common type of problem in mathematics that involves calculating the angles between the hands of a clock or determining the times when the hands align. It combines knowledge of angular motion and relative speed to set up equations that describe the positions of the hands over time.

Angular Velocity

This concept refers to the constant rate at which the clock’s hands move around the clock face. The minute hand and hour hand each rotate at their specific constant speeds, and understanding these speeds is essential for solving problems related to their positions at any given time.

Time Conversion

This concept includes converting continuous time measurements, often involving fractions of minutes, into more practical units like seconds. Rounding to the nearest second is necessary for interpreting the results in a real-world, easily understood format.

Transcript

00:01 Hello, everyone.

00:02 This is number 17 and is your geometry textbook.

00:06 This problem deals with angle measures in relation to time on a clock, along with a little bit of dimensional analysis for changing back and forth between the units.

00:19 Some of the shorthand i will use in this presentation is i will use mh to represent minute hand, hh to represent our hand, which means then that mis represents minute and h represents minute, and h represents.

00:31 Represents hours.

00:35 On the clock i have drawn over here on the left, the red pointer represents our minute hand, which is pointing to the 12.

00:46 And the black pointer represents the hour hand, which is pointing to the 12.

00:49 This represents the time of 12 o 'clock.

00:53 And our problem, we want to determine when will the hour hand and minute hand cross again after 12 o 'clock.

01:04 If we reason through it a little bit, we know that the minute hand fully goes around the entire clock every hour.

01:16 The clock is in the shape of a circle, so it's 360 degrees.

01:20 So we know that our minute hand rotates 360 degrees per hour.

01:33 I want to know how many degrees it goes per minute.

01:36 So there are 60 minutes in an hour, so i'm going to replace the bottom of this fraction with 60 minutes.

01:41 So 360 degrees in 60 minutes gives us 6 degrees per minute.

01:55 For our hour hand, it does not go, it does not rotate all the way around the circle every hour.

02:03 It takes 12 hours for the minute hand to get back to the 12 again.

02:07 So it only goes a fraction of the number of the degrees.

02:11 So there are 12 numbers evenly spaced around there.

02:15 So to go from 12 to 1 is 360 divided by 12, so that's 30 degrees.

02:22 So the hour hand rotates 30 degrees every hour.

02:30 And if we change this to minutes, this is 30 degrees in 60 minutes.

02:37 So it rotates 0 .5 degrees per minute.

02:45 When the hour hand and minute hand overlap, they are both pointing at the exact same point.

02:53 That means they have moved the exact same number of degrees.

02:57 So in the case of 12 o 'clock, both of them have completed a full 360 degrees, or they are now at zero degrees at the 12.

03:08 If they overlap again, they won't be at the 12.

03:12 They will be somewhere after the one, most likely since the minute hand fully circles, goes around the circle every hour, sometime within the one o 'clock hour, these two hands will overlap.

03:26 When they do overlap, the number of degrees that they have moved will be exactly the same.

03:33 That now allows us to start trying to create a formula to represent how many degrees each hand has moved on the clock at a specific time.

03:48 The minute hand is easy.

03:50 Every minute it moves six degrees.

03:53 So if it was 10 minutes, it moves 60 degrees.

03:56 30 plus 30, that means it's pointing to the two.

04:00 The hour hand is a little bit trickier.

04:03 It moves 0 .5 degrees per minute.

04:08 So it's going to matter how many hours have passed.

04:12 It's also going to matter how many minutes past the hour.

04:17 So for example, if we were at 2 .10, and we are trying to figure out how many degrees the hour hand has moved.

04:29 Well, it moves 0 .5 degrees per minute.

04:38 There's 60 minutes in an hour.

04:41 So if it's at 2 o 'clock, that's two hours that have passed.

04:47 And each of those are 60 minutes.

04:49 So we're going to do 60 times 2 to find out how many total minutes it has moved 0 .5 degrees plus the 10 minutes past, plus the 10 minutes past the hour.

05:10 That will give us how many degrees the hour hand moved, because it moves at 0 .5 degrees per minute.

05:18 So at two hours and 10 minutes, two hours have passed 60 minutes each hour, so 60 times two for the number of minutes there, plus 10 minutes past the hour.

05:32 So if we worked this problem out just to check and see if our thinking is correct so that we can write a formula.

05:45 In our calculators, we're going to do 0 .5 times parentheses, 60 times 2 plus 10.

05:54 And that's going to give us 65 degrees.

06:02 65 degrees makes sense because, the hour hand moves 30 degrees every hour.

06:10 So at two hours, it's moved 30 plus 30, 60 degrees...

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