Question

A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60cm, calculate the speed per hour with which the boy is cycling.

Answer 1

Hint: First of all, we will find the distance travelled by the wheel by multiplying the circumference to revolution and then, we will change the units of distance and time in km and hour respectively, by using the following conversion,
$$\begin{array}{l}1km={10}^{5}cm\\ 1\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}=60\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{s}\end{array}$$

Complete step-by-step answer:
We have been given that a boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. Also, the diameter of the wheel is 60cm, then, we have to find the speed per hour.
The diameter of a wheel is 60cm.
We know that radius is half of the diameter.
$$\Rightarrow Radius\left(r\right)={\displaystyle \frac{d}{2}}={\displaystyle \frac{60}{2}}=30cm$$
We know that, circumference of a circle having radius 'r' is $2\pi r.$
$$\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{m}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{e}\mathrm{l}\left(\mathrm{c}\right)=2\times \pi \times 30$$
Since, we know that $\pi ={\displaystyle \frac{22}{7}}$
$$\begin{array}{l}\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{m}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{e}\mathrm{l}\left(\mathrm{c}\right)=2\times {\displaystyle \frac{22}{7}}\times 30\\ \Rightarrow 188.57cm\end{array}$$
Now, the speed of the wheel will be equal to the product of revolution per second and circumference of the wheel.
$$\begin{array}{l}\Rightarrow Speed=140\times 188.57{\displaystyle \frac{cm}{minute}}\\ \Rightarrow 26399.8{\displaystyle \frac{cm}{minute}}\end{array}$$
Now, we have to find the speed in km per hour, so, we will have to convert cm into km and minute to hour by using the following conversion,
$$\begin{array}{l}1cm={10}^{-5}km\\ 1minute={\displaystyle \frac{1}{60}}hour\end{array}$$
$$\begin{array}{l}\Rightarrow Speed=23699.8\times {\displaystyle \frac{{10}^{-5}}{{\displaystyle \frac{1}{60}}}}\\ \Rightarrow 23699.8\times 60\times {10}^{-5}km/hr\\ \Rightarrow 1583988\times {10}^{-5}km/hr\\ \Rightarrow 15.84km/hr\left(\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\right)\end{array}$$
Therefore, the speed in km per hour with which the boy is cycling is equal to 15.84km/hr.

Note: Be careful while finding the value of circumference because by mistake we treat the given diameter as radius of the wheel which gives us a wrong value.
Also, remember that, after multiplying the revolution per second and circumference of the wheel, you will get the speed in cm/min not in km/hr. So, do not forget to change it.