at what times between noon and 1 p.m. are the hands of a clock perpendicular?

Answer 1

12:15 and 12:45

Answer 2

The first people are close, but not quite right.

By 12:15, the hour hand will have moved 1/4 of the way between 12 and 1, and by 12:45 it will have moved 3/4 of the way. Which means the times at which the hands are perpendicular are (approximately) 12:16 1/4. and 12:48 3/4.

Answer 3

12:15 and 12:45 are not exactly true. Take a clock and try setting the times. The angles will be a bit off from 90 degrees. 12:16 and 12:48 are the correct times.

Answer 4

Great question! (And wow, it looks like a lot of responders have forgotten that the hour hand moves too!)

One way to solve it is this: Pretend you've tied a string to the end of the hour hand, and you're suspending the whole clock by the string. In other words, the hour hand always points straight up, while the entire clock rotates backwords at a rate of 1 rev per 12 hours (i.e. 1/12 rev. per hour).

Now, the minute hand normally rotates clockwise at the rate of 1 rev per hour; but in this odd reference frame (with the clock itself rotating), the minute hand is actually rotating at:

1 rev/hr – 1/12 rev/hr = 11/12 rev/hr.

After it makes 1/4 revolution, the minute hand will be perpendicular to the hour hand. This takes this amount of time:

(1/4 rev) / (11/12 rev/hr) = 3/11 hr = 16 min, 22 sec. (approx.)

It'll be perpendicular again after it makes 3/4 revolution. This takes:

(3/4 rev) / (11/12 rev/hr) = 9/11 hr = 49 min, 5 sec. (approx.)

So, the perpendicular moments are at about 12:16:22, and 12:49:05

Answer 5

It is not 12:15 or 12:45 because the hour hand moves relative to what minute it is in that hour.
So we must set up an equation that states that the difference in degrees between the minute hand (6 degrees/ minute) and the hour hand (0.5 degrees/minute). At 12:00, they are parallel, therefor 6*m-0.5*m=90 or 5.5m=90 where m is the number of minutes passed.
m=16.363636... and the time is... 12:16:21.8181... p.m.
and the other time (since we know it is relatively close to 12:45 can be represented by 6*m-0.5*m=270.
5.5m=270
or m=49.090909... and the other time is 12:49:05.45454... pm.

Answer 6

let y be degrees for minutes
let x be degrees for hours

A complete revolution for the minutes hand is 12 times that for the hours hand.

==>

y=12x

we have two cases where the two hands are perpendicular:
y-x = 90
y-x = 270


1st case
------------
y-x = 90

replace y in the equation
==>

12x - x=90

11x = 90

==> x=90/11

==> y=12*90/11

60 minutes for y refer to 360 degrees
==> 1 minute of y refer to 6 degrees of y

==> y in minutes = (12*90/11)/6 = 16 minutes and 21 seconds and 0.818181816 of a second

time ~= 12:16:22

2nd case
-------------

y-x = 270

==>
12x - x = 270
11x = 270
x=270/11

==> y=12*270/11

1 minute of y refer to 6 degrees of y

==> y in minutes = (12*270/11)/6 = 49 minutes 5 seconds and 0.454545 of a second

time~= 12:49:5 (rounding 0.4545 to the nearest integer which is zero)

==> Times:
12:16:22
12:49:05

Answer 7

12:17 and 12:48 Because the hour hand moves while the minute hand is moving.

Answer 8

12 hours 16 minutes 21.8 seconds

Answer 9

TWO PERPENDICULAR LINES CROSS TO FORM A RIGHT ANGLE

EXAMPLE....

LOOK AT THE LETTER T.... THE ARE CROSSING TO FORM PERPENDICULAR LINES...

SO AN EXAMPLE OF PERPENDICULAR LINES WOULD BE

AT 12:15...

AT 12:15 THE LITTLE HAND IS AT 12 AND THE BING HAND IS AT 3... THOSE TWO LINES FORM A RIGHT ANGLE...

THE OTHER EXAMPLE IS 12:45

Answer 10

12:15 and 12:45