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

Answer:

$\frac{\pi }{30}$ 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 $\frac{1}{12}$

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.

∅=$\frac{2\pi }{12} = \frac{\pi }{6} rad$

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= $(4)\frac{\pi }{6} = \frac{2\pi }{3}$

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:

$\frac{60}{12} = 5$ minutes

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

The angle per minute is:

α= $\frac{2\pi/3 }{20} = \frac{2\pi }{(20)(3)} = \frac{\pi }{30}$ rad/min

Notice that the minimum angle is the angle mesured clockwise.