Respuesta :
Answer:
[tex]\frac{\pi }{30}[/tex] 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 [tex]\frac{1}{12}[/tex]
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.
∅=[tex]\frac{2\pi }{12} = \frac{\pi }{6} rad[/tex]
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= [tex](4)\frac{\pi }{6} = \frac{2\pi }{3}[/tex]
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:
[tex]\frac{60}{12} = 5[/tex] minutes
So, from the 12 mark to the 4 mark there are 20 minutes
The angle per minute is:
α= [tex]\frac{2\pi/3 }{20} = \frac{2\pi }{(20)(3)} = \frac{\pi }{30}[/tex] rad/min
Notice that the minimum angle is the angle mesured clockwise.
The concept of ratio and proportion is applied to solve the problem. In four hours, hours hand rotation is [tex]\dfrac{2\pi }{3}[/tex] radian.
What are ratio and proportion?
A ratio is an ordered couple of numbers a and b, written as a/b where b can not equal 0. A proportion is an equation in which two ratios are set equal to each other.
In 12 hours, the hour hand will rotate 2π radians.
[tex]\rm 12\ hours = 2\pi \\\\1\ \ \ \ hour = \dfrac{2\pi }{12}\\\\1 \ \ \ \ hour = \dfrac{\pi}{6}[/tex]
In one hour it rotates only [tex]\dfrac{\pi }{6}[/tex] radian.
Then in four hours, the rotation will be
[tex]\rm 1\ hour = \dfrac{\pi }{6}\\\\\rm 4\ hour = \dfrac{4* \pi }{6}\\\\\rm 4\ hour = \dfrac{2 \pi }{3}[/tex]
Thus, in four hours, hours hand rotation is [tex]\dfrac{2\pi }{3}[/tex] radian.
More about the ratio and proportion link is given below.
https://brainly.com/question/165414
