How many revolutions does the hoop make in one minute?

Question

Ursula and her friends are playing hoop rolling. Ursula rolls a large hoop on the ground. The hoop rotates through an angle of 3/2π radians in 1 second.

Answer 1

You can perform the steps in a few different orders, but they will all lead to the same place in the end. Here's the sequence I think you will find easiest:

First, the question asked for revolutions per minute, not second. Let's turn everything into minutes. 

I assumed you meant 3π

                                  2

for the angle, and not 3    , am I right? That gives you the complex fraction

                                 2π    

 

3πrad/2                                         Let's rewrite that as:

 1 second

 

3π rad   x   60 sec                          because dividing 3/2 by 1 is just 3/2, letting us reduce that complex fraction. You get

 2 sec          1 minute

 

90π rad                                            after cancelling the word "second" and reducing the 2 with 60.

 1 minute

 

Now, the sneaky but easy part. You complete 1 revolution for every 2π radians of spin, so we multiply two more fractions:

 

1 rev    x   90π rad                          Cancelling the word "radian" as well as reducing the π, the 2 and the 90 yields

 2π rad       1 minute

 

45 rev                                             Your book might also write it as "45 rpm".

1 minute

Answer 2

I'm guessing you meant (3/2)pi (=3pi/2) and not 3/(2pi).  First, compute the total number of radians that the hoop turns through in one minute (60 seconds):

 

(3pi/2 radians/sec)(60 sec) = 90pi radians

 

There are 2pi radians in one full revolution, so the number of revolutions in 90pi radians is:

 

(90pi radians)/(2pi radians/revolution) = _____? revolutions?