Question

airplane flying in a circular path with a radius of 2 miles at a rotational speed of 3 radians per minute. find the speed (in mph)?

Answer 1

you need to find out a few numbers:
1) how many radians is a full circle ? i.e. how many radians to equal 360º

Answer 2

there are 2pi radians in the circle
so 2pi=2 x 3.14 = 6.28

Answer 3

yes 2 pi radians in a circle
so we can figure out how many minutes it takes the plane to go 2 pi radians:
\[ \frac{2 \pi}{3} \text{ minutes}\]

next we need to know how far the plane traveled when it went around the circle once.
\[ C = 2\pi r = 2 \cdot 2 \pi \text{ miles} = 4 \pi \text{ miles}\]

Answer 4

distance = speed * time
and
speed = distance / time

that means we can find the speed in miles per minute
\[ speed = \frac{ 4 \pi}{\frac{2 \pi}{3}} = 4\pi \frac{3}{2\pi}= 6 \frac{ miles}{minute}\]

Answer 5

to change that to miles per hour multiply by 60 minutes per hour
\[ 6 \frac{miles}{\cancel{minute}} \cdot \frac{60 \cancel{minute}}{1 \text{ hour}} \]

Answer 6

so that will be 360 mph

Answer 7

Thank you so much for the detailed answer, I really appreciate your effort. thank you @phi