SOLUTION: What is the degree measure of the smallest angle between hour and minute hands of a clock at 11:10 pm. A)70 B)75 C)85 D)86

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Question 1144105: What is the degree measure of the smallest angle between hour and minute hands of a clock at 11:10 pm.
A)70
B)75
C)85
D)86
Found 3 solutions by Edwin McCravy, greenestamps, Alan3354:
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
We will analyze what happens during the 10 minute interval from 11:00 to 11:10.
 
The minute hand moves 12 times as fast as the hour hand, which is to say that
the hour hand moves only 1/12th as fast as the minute hand.

There are 12 numerals on a clock dial.  The whole clock dial is 360°.

Therefore the numerals on a clock dial are spaced 360°÷12 or 30° apart.

At exactly 11:00 the hour hand is on the 11 and the minute hand is on the 12.

At exactly 11:10 the minute hand is on the 2.  It has moved through 60° during
the 10 minutes from 11:00 to 11:10

The 11 numeral and the 2 numeral are 90° apart.

If the hour hand (for some strange reason!) had not moved during the 10 minute
period, then at 11:10 the hands would be 90° apart.  The answer would be 90°.

However the hour hand has moved 1/12 as much as the minute hand has moved.
The minute hand has moved 60° and the hour hand has moved 1/12 of 60° or 5°,
so the hands are 90°-5° or 85° apart at 11:10.

Answer: 85°

Edwin
Answer by greenestamps(13195)   (Show Source): You can put this solution on YOUR website!


There are many methods for solving this kind of problem. What works for me is to find the measure of each of the hands with respect to 12:00.

The hour hand moves 360 degrees in 12 hours, so its rate is 30 degrees per hour.

The minute hand moves 360 degrees in 60 minutes, so its rate is 6 degrees per minute.

At 11:10, the hour hand, measured from 12:00, has moved 30 degrees per hour for 11 1/6 hours, so 330+5 = 335 degrees.

At 11:10, the minute hand, measured from 12:00, has moved 6 degrees per minute for 10 minutes, so 60 degrees.

Then the larger angle between the two hands is 335-60 = 275 degrees; that makes the smaller angle 360-275 = 85 degrees.
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!

What is the degree measure of the smallest angle between hour and minute hands of a clock at 11:10 pm.
---------
The PM or AM is not relevant.
=================
A general solution:
Use angles measured from the 12 o'clock position clockwise (after all, it's a clock).
------
For a time H:M where H = hours and M = minutes:
The angle of the minute hand is (M/60)*360 = 6M degrees.
The angle of the hour hand is (H/12)*360 + 6M/12 degrees = 30H + M/2 degrees
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At 11:10, angle of the hour hand = 30*11 + 10/2 = 335 degs
Angle of the minute hand = 60 degs
H - M = 335 - 60 = 275 degs.
The smaller angle is 360 - 275 = 85 degrees

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