Question

The radius of a carriage wheel is 1 ft 9 inches, and in $\dfrac{1}{9}th$ of a second it turns through ${{80}^{{}^\circ }}$ about its centre, which is fixed. How many miles does a point on the rim of the wheel travel in one hour?

Answer 1

Hint:We will convert feet into inches as $1\text{ }ft=12\text{ }inch$. Also, we know that \[1\text{ }inch\text{ }=\text{ }2.54cm\]. We will first find the number of revolutions the wheel does in 1 second. Then, we know that \[1\text{ }hour\text{ }=\text{ }3600\text{ }seconds\]. So, we will find the distance travelled in 3600 seconds using the distance formula \[distance\text{ }=\text{ }speed\times time\].

Complete step-by-step answer:
It is given in the question that the radius of a carriage wheel is 1 ft 9 inch and $\dfrac{1}{9}th$ of a second it turns through ${{80}^{{}^\circ }}$ about its centre, which is fixed. Now, we have to find out how many miles a point on the rim of the wheel travels in one hour.
First, we will convert ft into inches. We know that $1\text{ }ft=12\text{ }inch$ and also we have to convert ft into inches. So, we have 12 inches in 1 ft. So, the radius of the wheel = $12+9\text{ }inches$
= $21\text{ }inches$.
Now, we know that \[1\text{ }inch\text{ }=\text{ }2.54cm\]. So, in 21 inches, we have = $21\times 2.54cm$
= $52.5cm$.
So, the radius of the wheel is 52.5 cm. We know that time taken by wheel to turn through ${{80}^{{}^\circ }}$= $\dfrac{1}{9}\text{seconds}$. So, in one second, wheel turns through = ?.
We will use a unitary method to find the degree that the wheel turn in 1 second. So, in $\dfrac{1}{9}\text{seconds}$ wheel rotates through ${{80}^{{}^\circ }}$. Now, 1 second it will rotate through ${{\left( 80\times 9 \right)}^{{}^\circ }}={{720}^{{}^\circ }}$.
Thus, in 1 second the wheel will rotate through 720 degrees.
Now, we know that in one complete round wheel rotates through 360 degree, it means the wheel rotates 2 times in 1 second as $\dfrac{{{720}^{{}^\circ }}}{{{360}^{{}^\circ }}}=2$ or, we can say that the wheel takes 2 revolution in 1 second.
Now, we know that \[1\text{ }hour\text{ }=\text{ }60\text{ }minutes\] and \[1\text{ }minute\text{ }=\text{ }60\text{ }seconds\], thus \[1\text{ }hour\text{ }=\left( 60\times 60 \right)\text{seconds=3600seconds}\]. Therefore, the number of revolution wheel takes 3600 seconds = $2\times 3600=7200\text{ }revolutions$.
Distance travelled by wheel in 1 revolution is circumference of the wheel = $2\pi r$, r being the radius of the wheel. So, the distance covered in 1 revolution = $2\pi \left( 52.5 \right)cm=105\pi cm$. Therefore distance covered in 7200 revolutions will be $105\pi \times 7200\text{ }cms=756000\pi \text{ }cms$.
Now, we know that \[1\text{ }km\text{ }=\text{ }100000\text{ }cm\], so \[756000\pi \text{ }cm=7.56\pi \text{ }km\], that is
$\left( 7.56\times 3.14 \right)kms=23.79384\text{ }km$. Now we will convert it into miles. We know that \[1\text{ }km\text{ }=\text{ }0.621\text{ }miles\], thus \[23.7389\text{ }km\text{ }=\text{ }\left( 0.621\times 23.7384 \right)\text{ }miles\]
= $14.75\text{ }miles$.
Therefore, the distance covered by a wheel in 1 hour is equal to 14.75 miles.

Note: Many students confuse the initial step of how to find the number of revolution wheel covers in 1 second. They may multiply $\dfrac{1}{9}$ with 60 to get the value of 1 second which is not correct at all. We have to use a unitary method to find the number of revolutions taken by the wheel in 1 second.