SOLUTION: Between 6 p.m. and 7 p.m. the hands of a clock make a ninety-degree angle on two occasions. If Jenny always leaves home to walk her dog when the first ninety-degree angle is formed

Question 1198138: Between 6 p.m. and 7 p.m. the hands of a clock make a ninety-degree angle on two occasions. If Jenny always leaves home to walk her dog when the first ninety-degree angle is formed, and arrives home when the second is formed, how much time, in hours, does Jenny spend walking her dog every week?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52784)   (Show Source): You can put this solution on YOUR website!
.
Between 6 p.m. and 7 p.m. the hands of a clock make a ninety-degree angle on two occasions.
If Jenny always leaves home to walk her dog when the first ninety-degree angle is formed,
and arrives home when the second is formed, how much time, in hours,
does Jenny spend walking her dog every week?
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We will measure angles starting from the position of the hour and the minute hands vertically up, at 12:00 (midday, the noon). The minute hand makes one full rotation in one hour, so its angular velocity is 360 degrees per hour, or 360/60 = 6 degrees per minute. The hour hand makes one full rotation in 12 hours, so its angular velocity is 360 degrees per 12 hours, or 360/12 = 30 degrees per hour = 30/60 = 0.5 degrees per minute. At 6:00 pm, the hour hand is in position 6*30 = 180 degrees from vertical position clockwise. The minute hand is in vertical position (= 0 degrees) at that time. "t" minutes after 6:00 pm, the hour hand is in position 180 + 0.5t degrees from vertical up; the minute hand is in position 6t degrees. After 6:00 pm, the hour hand and the minute hand make the right angle for the first time, when 180 + 0.5t = 6t + 90 degrees. From this equation, we find 180 - 90 = 6t - 0.5t, or 5.5t = 90, t = = 16.3636... minutes. After 6:00 pm, the hour hand and the minute hand make the right angle for the second time, when 6t = 180 + 0.5*t + 90 degrees. From this equation, we find 6t - 0.5t = 180 + 90, or 5.5t = 270, t = = 49.0909... minutes. ANSWER. Ian was out with his dog 49.0909 - 16.3636 = 32.7272 minutes = 32 minutes and 44 seconds (approximately.

Solved.

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Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

The time of day is not a necessary piece of information for solving the problem; the fact that the angles are 90 degrees at the beginning and end of each walk is sufficient.

Let t be the number of minutes that she spends walking her dog.

The hour hand moves 360 degrees in 12 hours, or 30 degrees per hour, or 0.5 degrees per minute; the minute hand moves 360 degrees in 1 hour, or 6 degrees per minute. So in t minutes the minute hand moves 6t degrees and the hour hand moves 0.5t degrees.

In the time that she is walking her dog, the minute hand goes from 90 degrees "behind" the hour hand to 90 degrees "ahead of" the hour hand, so the number of degrees the minute hand moves is 180 more than the number of degrees the hour hand moves:

= 32.73 minutes to the nearest hundredth

ANSWER: approximately 32.73 minutes per day; 229.09 minutes per 7-day week

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