Question

a use the applet to rotate abby clockwise from the 3 - oclock position to the 9 - oclock position. what measurement in radians do we use to describe the angle abby swept out on the ferris wheel?

b use the applet to rotate abby clockwise from the 9 - oclock position to the 12 - oclock position. what measurement in radians do we use to describe the angle abby swept out on the ferris wheel?

c use the applet to rotate abby clockwise from the 12 - oclock position to the 9 - oclock position. what measurement in radians do we use to describe the angle abby swept out on the ferris wheel?

Explanation:

Part a

Step1: Determine the fraction of the circle

A full circle is \(2\pi\) radians. Moving from 3 - o'clock to 9 - o'clock is half the circle (since 3 to 9 is 6 hours on a 12 - hour clock, which is \(\frac{6}{12}=\frac{1}{2}\) of the circle). But wait, clockwise rotation: from 3 to 9 clockwise is actually \(\frac{1}{2}\) of the circle? Wait, no. Wait, 3 - o'clock to 9 - o'clock clockwise: the angle between 3 and 9 on a clock (clockwise) is \(180^{\circ}\), but in radians, a full circle is \(2\pi\), so half a circle (180 degrees) is \(\pi\) radians? Wait, no, wait. Wait, 3 to 9 clockwise: the number of hours between 3 and 9 is 6 hours. Since a clock is a circle (360 degrees or \(2\pi\) radians) and 12 hours, each hour represents \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians per hour. So 6 hours would be \(6\times\frac{\pi}{6}=\pi\) radians? Wait, no, wait. Wait, when moving clockwise from 3 to 9, the angle swept is \(\pi\) radians? Wait, no, let's think again. The standard position: 3 - o'clock is along the positive x - axis (if we consider 3 - o'clock as (1,0) in polar coordinates). 9 - o'clock is along the negative x - axis. Moving clockwise from 3 to 9: the angle between positive x - axis (3 - o'clock) and negative x - axis (9 - o'clock) clockwise is \(\pi\) radians? Wait, no, counter - clockwise from 3 to 9 is \(\pi\) radians, but clockwise from 3 to 9 is also \(\pi\) radians? Wait, no, the total angle around a point is \(2\pi\). So clockwise from 3 to 9: the angle is \(\pi\) radians? Wait, no, let's calculate the number of hours. From 3 to 9 clockwise: 9 - 3 = 6 hours. Each hour is \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians (since 12 hours make a full circle of \(2\pi\) radians). So 6 hours: \(6\times\frac{\pi}{6}=\pi\) radians. Wait, but if we move clockwise, the angle should be negative? Wait, the problem says "measurement in radians". Maybe it's considering the magnitude? Wait, the Ferris wheel: when rotating clockwise, the angle swept. Let's see, the full circle is \(2\pi\) radians. From 3 to 9 clockwise: that's half the circle (since 3 to 9 is 6 positions out of 12, so \(\frac{6}{12}=\frac{1}{2}\) of the circle). So the angle is \(\pi\) radians (because \(\frac{1}{2}\times2\pi=\pi\)).

Step2: Confirm the calculation

Number of hours between 3 and 9: \(9 - 3=6\) hours.
Angle per hour: \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians per hour.
Total angle: \(6\times\frac{\pi}{6}=\pi\) radians.

Step1: Determine the fraction of the circle

From 9 - o'clock to 12 - o'clock clockwise. 9 to 12 is 3 hours. Each hour is \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians per hour.

Step2: Calculate the angle

Number of hours: \(12 - 9 = 3\) hours.
Angle: \(3\times\frac{\pi}{6}=\frac{\pi}{2}\) radians? Wait, no. Wait, 9 - o'clock to 12 - o'clock clockwise: 12 - 9 = 3 hours. Each hour is \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians. So 3 hours: \(3\times\frac{\pi}{6}=\frac{\pi}{2}\) radians? Wait, no, wait. Wait, 9 - o'clock is at \(\pi\) radians (if 3 - o'clock is 0 radians). 12 - o'clock is at \(\frac{3\pi}{2}\) radians? No, wait, let's set 3 - o'clock as 0 radians (positive x - axis). Then 12 - o'clock is \(\frac{3\pi}{2}\) radians (270 degrees), 9 - o'clock is \(\pi\) radians (180 degrees). Moving clockwise from 9 (\(\pi\)) to 12 (\(\frac{3\pi}{2}\)): the angle swept is \(\frac{3\pi}{2}-\pi=\frac{\pi}{2}\) radians? Wait, no, clockwise movement: from 9 to 12 clockwise, the angle is \(\frac{\pi}{2}\) radians? Wait, 9 to 12 is 3 hours, each hour is 30 degrees (\(\frac{\pi}{6}\) radians), so 3*30 = 90 degrees, which is \(\frac{\pi}{2}\) radians. Yes. So the angle swept is \(\frac{\pi}{2}\) radians? Wait, no, wait. Wait, when moving clockwise from 9 to 12, the number of hours is 3, so the angle is \(3\times\frac{\pi}{6}=\frac{\pi}{2}\) radians.

Step1: Calculate hours between 9 and 12

Hours from 9 to 12: \(12 - 9 = 3\) hours.

Step2: Calculate radians per hour

Radians per hour: \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians/hour.

Step3: Calculate total radians

Total radians: \(3\times\frac{\pi}{6}=\frac{\pi}{2}\) radians. Wait, but clockwise rotation, so the angle should be negative? But the problem says "measurement in radians", maybe it's the magnitude? Wait, the Ferris wheel: when rotating clockwise, the angle swept is \(\frac{\pi}{2}\) radians? Wait, no, let's check again. 9 - o'clock to 12 - o'clock clockwise: the angle between 9 and 12 on the clock (clockwise) is 90 degrees, which is \(\frac{\pi}{2}\) radians. So the answer is \(\frac{\pi}{2}\)? Wait, no, wait. Wait, 9 to 12 clockwise: the angle is \(\frac{\pi}{2}\) radians? Wait, 9 to 12 is 3 hours, each hour is \(\frac{\pi}{6}\) radians, so 3*\(\frac{\pi}{6}\)=\(\frac{\pi}{2}\) radians. Yes.

Step1: Determine hours between 12 and 9

From 12 - o'clock to 9 - o'clock clockwise: 12 to 9 clockwise is 9 hours? Wait, no. Wait, 12 to 9 clockwise: 9 hours? Wait, 12 to 9 clockwise: the number of hours is 9? No, 12 to 9 clockwise: 9 hours? Wait, 12 to 1 is 1 hour, 12 to 9 is 9 hours? No, that's counter - clockwise. Wait, clockwise from 12 to 9: 12 to 9 clockwise is 3 hours? No, wait, 12 to 9 clockwise: 12 -> 1 -> 2 ->...->9. That's 9 hours? No, that can't be. Wait, no, a clock is 12 hours. Clockwise from 12 to 9: the number of hours is 9? No, that's wrong. Wait, 12 to 9 clockwise: the angle between 12 and 9 clockwise is 270 degrees, which is \(\frac{3\pi}{2}\) radians? Wait, no. Wait, 12 - o'clock is at the top (if we consider 12 - o'clock as (0,1) in Cartesian coordinates). 9 - o'clock is at (- 1,0). Moving clockwise from 12 to 9: the angle swept is \(\frac{3\pi}{2}\) radians? Wait, no. Let's use the hour - per - radian method. Each hour is \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians. Clockwise from 12 to 9: the number of hours is 9? No, that's not right. Wait, 12 to 9 clockwise: 12 to 9 is 9 hours clockwise? No, that's counter - clockwise. Wait, no, clockwise from 12 to 9: 12 -> 11 -> 10 -> 9. That's 3 hours? No, 12 to 9 clockwise: 12 to 11 is 1, 11 to 10 is 2, 10 to 9 is 3. So 3 hours? No, that's 3 hours counter - clockwise. Wait, I'm confused. Wait, the clock: 12 - o'clock is 0 degrees (if we consider 12 - o'clock as the top, 3 - o'clock as 90 degrees, 6 - o'clock as 180 degrees, 9 - o'clock as 270 degrees). So moving clockwise from 12 (0 degrees) to 9 (270 degrees): the angle is 270 degrees, which is \(\frac{3\pi}{2}\) radians. But in terms of hours: clockwise from 12 to 9: the number of hours is 9? No, that's not. Wait, no, the correct way: the angle between 12 and 9 clockwise is 270 degrees, which is \(\frac{3\pi}{2}\) radians. But let's calculate using the hour method. Clockwise from 12 to 9: the number of hours is 9? No, that's incorrect. Wait, no, the total hours in a clock is 12. So clockwise from 12 to 9: the number of hours is 9? No, that's counter - clockwise. Wait, I think I made a mistake earlier. Let's re - define: the angle for clockwise rotation. Let's take 3 - o'clock as the positive x - axis (0 radians). Then 12 - o'clock is \(\frac{3\pi}{2}\) radians (270 degrees), 9 - o'clock is \(\pi\) radians (180 degrees)? No, that's not right. Wait, no: in standard position (mathematical), 0 radians is along the positive x - axis (3 - o'clock). Then 90 degrees ( \(\frac{\pi}{2}\) radians) is 12 - o'clock? No, no. Wait, no, 3 - o'clock is (1,0) in polar coordinates (0 radians). 12 - o'clock is (0,1) which is \(\frac{\pi}{2}\) radians. 9 - o'clock is (- 1,0) which is \(\pi\) radians. 6 - o'clock is (0, - 1) which is \(\frac{3\pi}{2}\) radians. Oh! I see my mistake earlier. So 3 - o'clock is 0 radians, 12 - o'clock is \(\frac{\pi}{2}\) radians (90 degrees), 9 - o'clock is \(\pi\) radians (180 degrees), 6 - o'clock is \(\frac{3\pi}{2}\) radians (270 degrees). So now, let's re - solve the problems.

Part a re - solved:

From 3 - o'clock (0 radians) to 9 - o'clock (\(\pi\) radians) clockwise. The angle swept is \(\pi - 0=\pi\) radians (since moving clockwise from 0 to \(\pi\) is \(\pi\) radians).

Part b re - solved:

From 9 - o'clock (\(\pi\) radians) to 12 - o'clock (\(\frac{\pi}{2}\) radians) clockwise. Wait, that can't be, because 12 - o'clock is \(\frac{\pi}{2}\) radians, which is less than \(\pi\) radians. So moving clockwise from \(\pi\) to \(\frac{\pi}{2}\) is a negative angle? Wait, no, in mathematics, clockwise…

Answer:

\(\pi\)

Part b