SOLUTION: Please help me solve this equation: The hour and minute of a watch are exactly together between 8 and 9 o'clock; required the time I don't understand how to work this problem

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Question 1117661: Please help me solve this equation:
The hour and minute of a watch are exactly together between 8 and 9 o'clock; required the time
I don't understand how to work this problem. I can't show my work because I wasn't able to work the problem at all.
I appreciate any help you can offer!
Found 4 solutions by josgarithmetic, ikleyn, solver91311, greenestamps:
Answer by josgarithmetic(39621)   (Show Source):
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Answer by ikleyn(52817)   (Show Source):
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.

I will count and compare the angles that the hour hand and the minute hand make with the direction "vertically up", which is their position at noon. The hour hand rotates 360 degs in 12 hours; hence, its angular velocity is = 0.5 degree per minute. The minute hand rotates = 6 degrees per minute; it is its angular velocity. Let the current time is 8 hours and t minutes, 0 <= t <= 60. The position of the hour hand is 8*30 + 0.5*t degrees (counting from the direction "vertically up")). The position of the minute hand is 6t degrees. The positions coincide - it gives you an equation 8*30 + 0.5t = 6t. Simplify and solve for t: 240 = 6t - 0.5t ====> 5.5t = 240 ====> t = = 43.6363 . . . minutes = 43 minutes and 38 seconds. Answer. The time is 8 hours 43 minutes and 38 seconds.

Solved.

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Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!

It takes the minute hand one hour to complete an entire circle, i.e. it is moving at 360 degrees per hour or 6 degrees per minute. Since there are 12 hours on the clock, and 360 divided by 12 is 30, the hour hand moves at a rate of 30 degrees per hour, or one-half degree per minute.

At exactly 8 o'clock, the hour hand points exactly to the 8 which is at a position of 240 degrees measured clockwise from 0 (i.e. 12). The minute hand points to 12, which is 0 degrees.

So the function that describes the motion of the hour hand during the interval from 8 to 9 o'clock is:

And the function that describes the motion of the minute hand during the interval from 8 to 9 is:

Where is the number of minutes past 8. When these two functions are equal, the hands point to the same place.

Solve for

Straight up, this will give you the number of minutes with a decimal fraction. If you want the answer in minutes and seconds, take just the decimal fraction part and multiply it by 60.

Don't forget to add the minutes to 8 o'clock to get the time of day this occurs. Alternatively, you can specify that your answer represents minutes after 8 o'clock.

John

My calculator said it, I believe it, that settles it

Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!

The responses from the other tutors show a perfectly good algebraic solution to your problem.

Here is another solution method that I like to use for this kind of problem.

The hour hand makes 1 revolution every 12 hours; the minute hand makes 12 revolutions every 12 hours.

So there are 11 times in every 12-hour period when the hour and minute hands are pointing in the same direction.

Since both hands move at constant rates, the times when the hands are pointing in the same direction will be separated by 12/11 hours.

The two hands are pointing in the same direction at 12 o'clock.

The time between 8 and 9 o'clock when the hands are pointing in the same direction is 8*(12/11) hours after 12 o'clock.

8*(12/11) hours = 8 and 8/11 hours
8/11 hours = 480/11 minutes = 43 7/11 minutes
7/11 minutes = 420/11 seconds = 38 2/11 seconds

The two hands are pointing in the same direction at 8:43:38 2/11