MHB Distance Traveled by Clock Hour Hand in 1hr 20min

[paulmdrdo1]

if the hour hand of a clock has a length of 4 in. how far does its tip travel in 1hr and 20min?

 

[Opalg]

if the hour hand of a clock has a length of 4 in. how far does its tip travel in 1hr and 20min?

The tip of the hour hand travels round a circle of radius 4 in, covering a complete revolution in 12 hours. You know the formula for the circumference of the circle. So what fraction of that will be be covered in 1hr 20 min?

 

[paulmdrdo]

you can also use ratio-proportion to solve the problem.

$\displaystyle\frac{2\pi}{12\text{ hr}}=\frac{a}{1\text{ hr and }20\text{ min}}$

"a" is the angle in radian that would be generated in time 1hr and 20 min. - convert the minute part into hour.

 

[paulmdrdo1]

but how do you know when to use that technique? markfl? latebloomer? and i didn't understand how you came up with that equation $\displaystyle\frac{2\pi}{12\text{ hr}}=\frac{a}{1\text{ hr and }20\text{ min}}$

can you please explain further. anyone who's online please i want an urgent answer. thanks!

 

[MarkFL]

I would use the formula for length of the circular arc (essentially what Opalg is hinting at):

$$s=r\theta$$

We are given the radius $r=4\text{ in}$, and the angle $\theta$ can be determined from the elapsed time. The hour hand makes a complete revolution in 12 hours, and a complete revolution is $2\pi$ radians. What fraction of 12 hours is 1 hour and 20 minutes? When you find this fraction, which represents the fraction of a complete revolution the hour hand makes, then multiply this fraction by the complete revolution to find the angle through which the hour hand turns in the given time.

So, how many hours is 1 hour and 20 minutes?

 

[paulmdrdo1]

1hr and 20 is 4/3 hr. 12X4/3 = 16 this is my understanding of "what fraction of 12hrs is 1hr and 20 min". is this right?

 

[MarkFL]

Yes, 1 hour 20 minutes is 4/3 hour, but to find what fraction this is of 12, we want to divide not multiply. For example, we know 6 hours is 1/2 of 12 hours, and this can be found from 6/12 = 1/2. :D

 

[paulmdrdo1]

4/3/12 = 16 is this right?

 

[Opalg]

4/3/12 = 16 is this right?

No. 4/3 divided by 12 is $\dfrac4{3\times12}$.

 

[MarkFL]

4/3/12 = 16 is this right?

No, we want:

$$\frac{4/3}{12}=\frac{1/3}{3}=\frac{1}{3\cdot3}$$

We know that 4/3 is smaller than 12, so when we divide 4/3 by 12, we should expect to get a fraction smaller than one.

 

[paulmdrdo1]

oh my it's 1/9. now if we plugged it in $s = r\theta$ i will have $s=4(\frac{1}{9})(2\pi)=2.792\, in$ it seems correct now.

but i have a follow up question why is ratio and proportion also works in this problem?

 

[MarkFL]

Unless you are required to use a decimal approximation, I would leave the answer exact:

$$s=\frac{8\pi}{9}\text{ in}$$

Using proportions, which is quite similar to what we've just done, we may state in words:

12 hours is to one revolution what 4/3 hours is to some part of a revolution. Stated mathematically, this is:

$$\frac{12\text{ hr}}{2\pi}=\frac{\frac{4}{3}\text{ hr}}{\theta}$$