Radians distance traveled.

[Heschel6]

I answered this question and it was marked wrong and I can't figure out why.

Q: "The minute hand on a watch is .4 inches long. Write an expression to determine how far in inches does the tip of a minute hand travel as the hand turns through 60(deg)."


A: I answered .4(11pi/6)

I was told the correct answer was .4(pi/3) but I figured that the hand on a watch moves clock-wise so to travel through 60 deg it would have to go -300(deg)

 

[pka]

I answered this question and it was marked wrong and I can't figure out why.

Q: "The minute hand on a watch is .4 inches long. Write an expression to determine how far in inches does the tip of a minute hand travel as the hand turns through 60(deg)."
I was told the correct answer was .4(pi/3) CORRECT.

\(\displaystyle \mathfrak{M}=\mathcal{r}\theta\) is the formula for the length \(\displaystyle \mathfrak{M}\) of the arc subtended by an angle \(\displaystyle \theta\) in a circle of radius \(\displaystyle \mathcal{r}\).

 

[Heschel6]

\(\displaystyle \mathfrak{M}=\mathcal{r}\theta\) is the formula for the length \(\displaystyle \mathfrak{M}\) of the arc subtended by an angle \(\displaystyle \theta\) in a circle of radius \(\displaystyle \mathcal{r}\).

Thanks for your help

 

[Ishuda]

I answered this question and it was marked wrong and I can't figure out why.

Q: "The minute hand on a watch is .4 inches long. Write an expression to determine how far in inches does the tip of a minute hand travel as the hand turns through 60(deg)."


A: I answered .4(11pi/6)

I was told the correct answer was .4(pi/3) but I figured that the hand on a watch moves clock-wise so to travel through 60 deg it would have to go -300(deg)

Making the assumption that the clock hand started at the 0\(\displaystyle ^\circ\) mark along the positive x axis and the 60\(\displaystyle ^\circ\) mark was being measured going counter-clockwise, you would (almost) be correct [it would travel through \(\displaystyle 2\, \pi\, -\, \frac{\pi}{3}\, =\, \frac{5\, \pi}{3}\) radians and the distance would be \(\displaystyle 0.4\, (\frac{5\, \pi}{3})\) inches].

However, the question was "turns through 60(deg)" which is a different story. It travel through 60\(\displaystyle ^\circ\), whether going clockwise or counter-clockwise, when it has traveled \(\displaystyle \frac{\pi}{3}\) radians.

 

[pka]

Making the assumption that the clock hand started at the 0\(\displaystyle ^\circ\) mark along the positive x axis and the 60\(\displaystyle ^\circ\) mark was being measured going counter-clockwise, you would (almost) be correct [it would travel through \(\displaystyle 2\, \pi\, -\, \frac{\pi}{3}\, =\, \frac{5\, \pi}{3}\) radians and the distance would be \(\displaystyle 0.4\, (\frac{5\, \pi}{3})\) inches].

However, the question was "turns through 60(deg)" which is a different story. It travel through 60\(\displaystyle ^\circ\), whether going clockwise or counter-clockwise, when it has traveled \(\displaystyle \frac{\pi}{3}\) radians.

That is nonsense. Measures of arcs have absolutely nothing to do with clock-wise or counter-clock-wise.

The measure of any arc is simply the product of the radius times the measure of the subtending angle.

In fact that is the derivative of the definition of one radian.

The measure of the angle subtending an arc of length equal to the radius is one radian.

 

[Ishuda]

Making the assumption that the clock hand started at the 0\(\displaystyle ^\circ\) mark along the positive x axis and the 60\(\displaystyle ^\circ\) mark was being measured going counter-clockwise, you would (almost) be correct [it would travel through \(\displaystyle 2\, \pi\, -\, \frac{\pi}{3}\, =\, \frac{5\, \pi}{3}\) radians and the distance would be \(\displaystyle 0.4\, (\frac{5\, \pi}{3})\) inches].

However, the question was "turns through 60(deg)" which is a different story. It travel through 60\(\displaystyle ^\circ\), whether going clockwise or counter-clockwise, when it has traveled \(\displaystyle \frac{\pi}{3}\) radians.

That is nonsense. Measures of arcs have absolutely nothing to do with clock-wise or counter-clock-wise.

The measure of any arc is simply the product of the radius times the measure of the subtending angle.

In fact that is the derivative of the definition of one radian.

The measure of the angle subtending an arc of length equal to the radius is one radian.

Pka,

Yes, it is true that the measure of an arc is the product of the radius and the measure of the subtending angle. However, the length of the arc [or, if you prefer, the subtending angle} depends on which way you go, counter clockwise or clockwise. If you have a circle defined by
x² + y² = 1
and two points A=(1,0) and another point B=(0,1), the length of the arc is not the same whether you go counter-clockwise or whether you go clockwise. If you go from A to B counter-clockwise, the length is \(\displaystyle \frac{\pi}{2}\) but if you go from A to B clockwise the length is \(\displaystyle \frac{3\, \pi}{2}\). If it didn't make a difference then we would have 3=1 and I don't think that is quite right [unless you are counting in binary which I don't believe we are].