Question

At what time between 7:00 pm and 8:00 pm will the minute hand be 90 degrees behind the hour hand?

Answer 1

??

Answer 2

@geoffb are you typing a reply?

Answer 3

7 pm is at 210 degrees.

Find out how far the hour and minute hands move (both in terms of time, t), and plug those into the following formula.

210 degrees + distance hour hand travels - distance minute hand travels = 90 degrees

You should be left with one variable to solve for, t.

Answer 4

7:30?

Answer 5

Nope. It will be more in-depth. What are your formulas for distance hour hand travels and distance minute hand travels, both in terms of t?

Answer 6

7;45?

Answer 7

No. Show your work, because you're clearly just guessing.

Answer 8

I am not understanding

Answer 9

Look at the clock. It's 360 degrees all around, so 90 degrees is one fourth of that, or 15 minutes(three "numbers" back). You do actually have to show work, but remember that the hour hand also moves forward

Answer 10

Exactly as LT said. Very simple deduction will tell you that the minute hand will have to be between 20 and 25 minutes.

Answer 11

my bad, 6:15

Answer 12

Is the answer 7:15

Answer 13

@Mouser: no, because the hour hand doesn't stay on the hour. It moves fractionally.

No, it's not 7:15. Did you read what LT and I said? It has to be between 7:20 and 7:25.

Answer 14

I'll try to show you another example with the same reasoning. See if you can apply it to your own question.

Let's say I want to know when, between 1 and 2, the hour hand and minute hand form a straight line (that is, they're 180 degrees apart). The very first thing you should ask is "what is reasonable?"

We know that the hour hand will point somewhere between 1 and 2. Therefore, the minute hand *must* point between 7 and 8 in order to satisfy the conditions ([1] hour hand between 1 and 2, and [2] minute hand is 180 degrees from hour hand).

There are 360 degrees on a clock. Since there are 12 hours, each hour is 360/12 = 30 degrees.

In an hour, the minute hand does a full revolution (it moves 360 degrees). Therefore, it moves 360 degrees/60 minutes, or 6 degrees per minute.

In an hour, the hour hand moves 1/12 of a revolution. It therefore moves 360 degrees/(12 x 60 minutes), or 0.5 degrees per minute.

The hour hand will move at a rate of r1.

The minute hand will move at a rate of r2.

Let t be the time in minutes until the hands are 180 degrees apart.

At 1 pm, the minute hand is 1/12 of the clock behind the hour hand (-30 degrees).

So, you can say that, starting 30 degrees behind, subtract the rate of movement of the hour hand from the rate of movement of the minute hand to get 180 degrees, your desired amount of separation.

\[-30 + r_{2} - r_{1} = 180\]
\[r_{2} - r_{1} = 210\]

You know that r2 = 6t and r1 = 0.5t (or t/2)

\[6t - \frac{t}{2} = 210\]
\[12t - t = 420\]
\[11t = 420\]
\[t = 38.18\]

Multiply 0.18 minutes by 60 seconds/minute to get 10.8 seconds (we'll say 11).

Between 1 and 2, the hour and minute hands will form a straight line at 38 minutes and 11 seconds past the hour.