Tammy is at the dentist's office waiting on her appointment. She notices that the 6-inch-long minute hand is rotating around the
clock and marking off time like degrees on a unit circle. Part 1: How many radians does the minute hand move from 1:20 to 1:55? (Hint: Find the number of degrees per minute first.)
Part 2: How far does the tip of the minute hand travel during that time?
Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 5π inches?
Part 4: What is the coordinate point associated with this radian measure?
In order to find how many radians the minute hand moves from 1:20 to 1:55, we need to remember that there are 60 minutes in an hour (clock) and there are 360 degrees in the clock since the clock is a circle. After dividing 360 by 60, we find that each minute is equal to 6 degrees. After that, we can subtract the times, which tells us that there are 35 minutes between 1:20 and 1:55. Using this we can just multiply this out, to get 35 times 6, which is equal to 210 degrees. We can get our final answer by converting this into degrees. Since one 1 degree is about 0.0174, we can set up a proportion. After solving, we will get that the minutes hand moves 3.555 radians in total.
Part 2:
In order to find how much the minute hand moves, we must find the circumference, so we get c= pi times diameter. Once plugging in the 12, we see that c=37.68. 37.68 is the circumference of the entire clock and since we only need the circumference/length/distance of 35 minutes, we can set up the proportion of 37.68 in./60=x/35 and solve to get 21.98, which means 21.98 is how far the minute hand travels in 35 minutes.
