Hi, there--
THE PROBLEM:
A car has wheels which are 140 cm in diameter. How many complete revolutions per minute
does each wheel make when the car is running at a speed of 66 km/hr?
A SOLUTION:
This is a unit conversion problem. We will work our way through a series of conversions to get
to the desired information. We will need to keep close eye on the units of our answers as we
proceed.
We are asked to find the number of complete revolutions per minute. In other words, how
many times will the wheel make a complete turn in a one-minute period. Let's start with what
we know:
The wheels have a diameter of 140 cm. In order to find the length of one revolution of the
wheel, we need to know the circumference C of the wheel. The formula for the circumference
of a circle is "the diameter times [pi]." Estimate [pi] as 3.14.
C = [140 cm] * [3.1416] = 439.82 cm
So, each time the wheels make one revolution the car travels 439.82 cm. We can write this as
439.82 cm per revolution, or 439.82 cm/rev.
We know that the car is traveling as 66 km per hour. Remember we want to know the number
of revolutions in one minute. Let's translate 66 km/hr in km per minute.
Since the car is traveling at 66 km per hour that means it travels 66 km in one hour. There are
60 minutes in one hour, so the car travels 66 km in 60 minutes. We want to know how far the
car travels in one of those minutes, so we divide:
[66 km/hr] / [60 min/hr] = 1.1 km/min
The car travels 1.1 km per minute.
Notice that our car speed is expressed in km, but or wheel revolutions are expressed in cm.
We need to translate one of those rate. Let's rewrite the car speed in cm/min. We will do this
in two steps for the sake of clarity. There are 1000 meters (m) in one km. We want to know
how many meters there are in 1.1 km, so we multiply:
[1.1 km/min] * [1000 m/km] = 1100 m/min.
We repeat this process to convert meters to cm. There are 100 cm in one meter. We want to
know how many centimeters there are in 1100 m, so we multiply,
[1100 m/min] * [100 cm/min] = [110000 cm/min]
Now we know that a car traveling at a constant speed of 66 km per hour, travels 110000 cm
each minute. We also know that the car travels 439.82 cm for each revolution of the wheel.
We would like to know how many of those revolutions will fit in 110000 cm, so we divide
[110000 cm/min] / [439.82 cm/rev] = [250.10 rev/min]
The wheels make 250.10 revolutions each minute. That's 250 complete revolutions, plus a
little bit more.
As you get better at these unit conversion problems, it's common to see the problem laid out
in a long string of conversions, sometimes called dimensional analysis. Here's what that looks
like for your problem:
[66 km/hr] / [60 min/hr] * [1000 m/km] * [100 cm/m] / [439.82 cm/rev = [250.10 rev/min]
Hope this helps! Feel free to email me with a follow-up question if you have questions about
the explanation.
Mrs. Figgy
