Calculating the positions of the hands (2)?

Question

The hands of a watch are at an exactly equal angle to the vertical axis throug the pivot point of the hands, like arms opened invitingly to the buyer.
a)What is the exact time on this watch?

The hands are symmetrical with regard to the horizontal axis through the pivot point of the hands and the little hand is near the 8.
b) What is the exact time?

Answer 1

We'll use a similar approach as to your other question.
As before, the big hand moves at a rate of 6 degrees per minute, and the little hand at 0.5 degrees per minute.
So t minutes after midnight, the big hand has moved 6t degrees, and the little hand has moved 0.5t degrees. Now, we want the two angles to add up to a multiple of 360 degrees to make them in the required positions.
So we want 6.5t = multiple of 360 degrees. So the required positions occurs every 360/6.5 minutes from midnight (lots of answers).

Note that that was basically the same question as your last one - if the hands were aligned when one was moving the wrong way; if it was actually moving the right way, they will be perfectly symmetrical as here. So thats why the answers are the same.

For the second question, its very similar. This time we want the two angles to add to 180 more than a multiple of 360.
So 6.5t + 180 is a multiple of 360.
Now, since the little hand is near the 8, we know that t (which is in minutes since midnight) should be approximately 8 hours * 60 = 480 minutes. 6.5 * 480 + 180 is 3300, and the nearest multiple of 360 is 3240. So t = (3240-180)/6.5 = 470.769 minutes since midnight.
That works out to be about 7:50:46, to the nearest second.

Answer 2

a) The times when the hands are at equal angles from the vertical (12-to-6) are:
11 + 30/6.5 = 11:04:36.9
10 + 60/6.5 = 10:09:13.8
9 + 90/6.5 = 9:13:50.8 ---- closest to straight line
8 + 120/6.5= 8:18:27.7
7 + 15/6.5 = 7:23:04.6
6 + 180/6.5 = 6:27:41.5
b)

6t - 270 = +60 - 0.5t
6.5t = 330
t = 7:50:46.2

Answer 3

a. 9 minutes of 2:00.

b. 20 minutes of 4:00.

Answer 4

3:15am