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# Two wheels are connected via a conveyor belt. The larger

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Two wheels are connected via a conveyor belt. The larger [#permalink]  02 Jun 2012, 21:43
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Two wheels are connected via a conveyor belt. The larger wheel has a 40cm diameter and the smaller wheel has a 32cm diameter. In order for the conveyor belt to work smoothly, each wheel must rotate the exact same number of centimetres per minute. If the larger wheel makes r revolution per minute, how many revolutions does the smaller wheel make per hour in terms of r?

A. 1280π
B. 75r
C. 48r
D. 24r
E. (64π)/3
[Reveal] Spoiler: OA
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  16 Jul 2013, 16:24
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macjas wrote:
Two wheels are connected via a conveyor belt. The larger wheel has a 40cm diameter and the smaller wheel has a 32cm diameter. In order for the conveyor belt to work smoothly, each wheel must rotate the exact same number of centimetres per minute. If the larger wheel makes r revolution per minute, how many revolutions does the smaller wheel make per hour in terms of r?

A. 1280π
B. 75r
C. 48r
D. 24r
E. (64π)/3

1. Let the number of revolutions made by the smaller wheel in 1 hr be s
2. Given: Number of centimeters rotated by both the wheels in 1 hr is the same

3. Number of centimeters rotated by the larger wheel in one revolution = its circumference=2*pi*20=40Pi
4. Number of centimeters rotated by the larger wheel in 1 hr = 40*pi*60*r

5. Number of centimeters rotated by the smaller wheel in one revolution=its circumference=2*pi*16=32*pi
6.Number of centimeters rotated by the smaller wheel in 1 hr= 32*pi*s

7. Equating (4) and (6) based on (2),

40*pi*60*r = 32*pi*s
i.e., s= 75r
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  03 Jun 2012, 02:16
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Expert's post
macjas wrote:
Two wheels are connected via a conveyor belt. The larger wheel has a 40cm diameter and the smaller wheel has a 32cm diameter. In order for the conveyor belt to work smoothly, each wheel must rotate the exact same number of centimetres per minute. If the larger wheel makes r revolution per minute, how many revolutions does the smaller wheel make per hour in terms of r?

A. 1280π
B. 75r
C. 48r
D. 24r
E. (64π)/3

The radius of the larger wheel is 20cm so, its circumference is $$2\pi{R}=40\pi$$. Since the larger wheel makes r revolution per minute or 60r revolutions per hour then it rotates $$60r*40\pi$$ centimeters per hour.

The radius of the smaller wheel is 16cm so, its circumference is $$2\pi{R'}=32\pi$$. Now, in order the smaller wheel to rotate the same centimeters per time interval it should make $$\frac{60r*40\pi}{32\pi}=75r$$ revolutions per hour.

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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  25 Sep 2013, 22:33
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Visualization is the key in such problems.

To work smoothly

The belt left by big wheel should be taken up by small wheel. Implicitly,

Length left by big wheel /min = length accepted by small wheel/min

40 *pi * r = 32 * pi * x

x = (5/4) * r in minute

x= (5/4) * r * 60 in hour
Hence B
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  11 Jul 2012, 09:39
Hai guys..

I just tried this way..

40P/no of centimeters = R per minute

So no of centi = 40P/R

since both the wheels move at the same rate i.e in terms of centi
Revol for the smaller one = 32P*R/40P
=4/5R per minute and 48R per hour...

What am I missing ?
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  12 Jul 2012, 01:51
My take on it,
the wheel will cover PI*40 cm in one rotation. Thus to if we know total distance in a unit time, the expression for rev. should be
No of centi per minute/PI*40 = rev per minute
Also
as if you look at the unit at both side
LHS:
no of cm per minute => units are cm/min
PI*40 => units are cm
RHS
Rev per minute => units are (min)^-1 or (1/min)
Thus,
[cm/min] /[cm] = [1/min]
showing the equation balances
and in your case you inverted the LHS of the equation.
Rest is you can equate the total distance for both wheel
PI*32*R = PI*40 *r
R = 5/4*r in cm/m or 5/4*r*60 = 75*r cm/h.
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  12 Jul 2012, 02:15
Expert's post
Priyanka2011 wrote:
Hai guys..

I just tried this way..

40P/no of centimeters = R per minute

So no of centi = 40P/R

since both the wheels move at the same rate i.e in terms of centi
Revol for the smaller one = 32P*R/40P
=4/5R per minute and 48R per hour...

What am I missing ?

Check this: two-wheels-are-connected-via-a-conveyor-belt-the-larger-133697.html#p1092048

The circumference of the larger wheel is $$2\pi{R}=40\pi$$. Since the larger wheel makes $$r$$ revolution per minute or $$60r$$ revolutions per hour, then it rotates $$60r*40\pi$$ centimeters per hour.

The radius of the smaller wheel is 16cm so, its circumference is $$2\pi{R'}=32\pi$$. Now, in order the smaller wheel to rotate the same centimeters per time interval it should make $$\frac{60r*40\pi}{32\pi}=75r$$ revolutions per hour.

Hope it's clear.
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  15 Jul 2013, 23:14
Expert's post
Bumping for review and further discussion.
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  24 Jul 2013, 05:53
I just applied the values:

r revolutions in 1 min, so 60r in 60mins.

Consider a point on this wheel. In each revolution of the larger wheel, this point travels 40 cms, so at the end of 60r it travels 2400cms.

How many revolutions will it take for a point on the smaller wheel to cover this distance? 2400/32 = 75r
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  27 Nov 2013, 03:42
There is a simple formula for this kind of problems. Given two pulleys/wheels are connected tightly with a conveyor belt, each wheels completes same distance in an hour as the other wheel unless there is slippage. only their revolutions per min or hour differ due to their difference is size.

If r1 and r2 are revolutions per hour of each wheel and
R1 and R2 are radius of each wheel.

equating distance of each wheel

$$2pi*R1*r1 = 2pi*R2*r2$$

$$R1*r1 = R2*r2$$

$$\frac{R1}{R2} = \frac{r2}{r1}$$

given 3 of the values we can find the 4th one.
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  12 Dec 2013, 11:53
Two wheels are connected via a conveyor belt. The larger wheel has a 40cm diameter and the smaller wheel has a 32cm diameter. In order for the conveyor belt to work smoothly, each wheel must rotate the exact same number of centimetres per minute. If the larger wheel makes r revolution per minute, how many revolutions does the smaller wheel make per hour in terms of r?

Interesting to note that the larger wheel has a diameter of 40 (8*5) while the smaller one has a diameter of 32 (8*4)...

If the large wheel has a diameter of 40 and the small wheel, 32, then their circumferences are 40pi and 32pi respectively. In order for them to move the conveyor belt at the same rate, the smaller wheel would need to rotate 1.25 times as fast as the larger wheel. Lets say the large wheel makes 10 revolutions per minute, the smaller wheel would need to make 10*1.25 = 12.5 revolutions per minute. If the large wheel makes 10 revolutions per minute it makes 600 per hour. Therefore, the smaller wheel would need to make 600*1.25 = 750 revolutions per hour.

If r = 10, then the answer choice must be b.

B. 75r
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Re: Two wheels are connected via a conveyor belt. The larger [#permalink]  12 Dec 2014, 21:00
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Re: Two wheels are connected via a conveyor belt. The larger   [#permalink] 12 Dec 2014, 21:00
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