Question

Suppose theta(????) measures the minimum angle between a clock’s minute and hour hands in radians. What is theta′(????) at 4 o’clock? Give your answer in radians per minute. (Use symbolic notation and fractions where needed.)

Answer 1

Theta'(t) represents the rate of change of the angle between the clock hands at 4 o'clock in radians per minute. It is calculated as the difference in angular velocities of the minute and hour hands, resulting in 32π/1080 radians per minute.

Step-by-step explanation:

To find theta'(t) at 4 o'clock, we look at the rate of change of the minimal angle between the hour and minute hands of the clock in radians per minute. Let's note that the hour hand moves 1/12 of a full revolution (2π radians) every hour, or (1/12)*(2π)/60 radians per minute. At 4 o'clock, the hour hand is 1/3 of the way through its journey from 3 to 6 o'clock. Thus, the hour hand will rotate an angle of (1/3)*(1/12)*(2π)/60 radians per minute.

The minute hand moves with a constant speed of 2π radians per 60 minutes, which simplifies to π/30 radians per minute. The rate change of the angle, theta', is then the difference between the angular velocities of the minute and hour hands. Hence, theta'(t) at 4 o'clock is π/30 - (1/3)*(1/12)*(2π)/60, which simplifies to 11π/360 - π/1080 radians per minute, or 32π/1080 radians per minute after finding a common denominator and combining the terms.

Answer 2

`(\pi )/(30)` radians per minute.

Step-by-step explanation:

In order to solve the problem you can use the fact that the angle in radians of a circumference is 2π rad.

The clock can be seen as a circumference divided in 12 equal pieces (because of the hour divisions). Each portion is
`(1)/(12)`

So, you have to calculate the angle between each consecutive hour (Let ∅ represent it). It can be calculated dividing the angle of the entire circumference by 12.

∅=
`(2\pi )/(12) = (\pi )/(6) rad`

Now, you have to find how many pieces of the circumference are between 12 and 4 to calculate the angle (Because 4 o'clock is when the minute hand is in 12 and the hour hand is in 4)

There are 4 portions from 12 to 4, so the angle (Let α represent it) is:

α=
`(4)(\pi )/(6) = (2\pi )/(3)`

But the answer is asked in radians per minute. So you have to divide the angle by the amount of minutes between the hands of the clock at 4 o'clock.

There are 60 divisions in a clock for representing minutes, therefore in every portion there are:

`(60)/(12) = 5` minutes

So, from the 12 mark to the 4 mark there are 20 minutes

The angle per minute is:

α=
`(2\pi/3 )/(20) = (2\pi )/((20)(3)) = (\pi )/(30)` rad/min

Notice that the minimum angle is the angle mesured clockwise.