Clock problem involving angles
I am studying for a geometry test and am having trouble with a review problem at the end of the chapter. This is not homework.
One problem asks the following: At 3:00, the hands of a clock form an angle of 90 degrees. To the nearest second, at what time will the hands of the clock next form a 90 degree angle? I figure that 4:05 is one time after three that the hands will form a 90 degree angle. So I figure that halfway betwee 3:00 and 4:05, the hands will form a right angle. Since half of 65 min is 32.5 minutes, so the answer should be 3:32:30. However, the answer at the end of the book is 3:32:44??? Don’t understand this. The second part of the problem is: To the nearest second, what is the first time after 2:00 that the hands of a clock will form an angle 2.5 times as great as the angle formed at 2:00? The angle formed at 2 is 60 degrees. So we need the next time to form a 150 degree angle. The answer is 2:38:11. Not sure how to approach this.