Question

The diameter of a wheel of a bus is 90 cm which makes 315 revolutions per minute. Determine its speed in kilometers per hour. $\left( Use\ \pi =\dfrac{22}{7} \right)$

Answer 1

Hint: We will be using the concepts of circles to solve the problem. We will also be using the fact that a wheel covers a distance equal to its circumference in one rotation. Then multiplying it with number of revolutions we get the total distance travelled by bus. Use formula $\text{speed}\ \text{=}\dfrac{\text{distance}}{\text{time}}$ to calculate speed and later convert it into kilometers per hour.

Complete step-by-step answer:
Now, we have been given that the diameter of the wheel is 90 cm and also that its makes 315 revolutions per minute. We have to find the speed of the bus in kilometers per hour.
Now, we know that the distance covered by a wheel in one rotation is equal to its circumference. Also, the circumference of the circle is $2\pi r$ where r is the radius of the circle.
So, we have the distance covered by wheel in 315 revolution,
$\begin{align}
  & =315\times 2\times \pi \times r \\
 & =315\times 2\times \pi \times \dfrac{90}{2}cm \\
 & =315\times \pi \times 90\ cm \\
\end{align}$
Now, we know that,
$\text{speed}\ \text{=}\dfrac{\text{distance}}{\text{time}}$
So, we have the distance of the wheel of the bus covered in 1 min. So, the corresponding speed is,
$\text{speed}\ \text{=}\dfrac{315\times \pi \times 90cm}{1\ \min }$
Now, we have to convert cm to km and min to hours to find the speed in km/hr. Also, we know that,
$\begin{align}
  & 1km={{10}^{5}}cm \\
 & 1hr=60\min \\
 & \Rightarrow \dfrac{1}{60}hr=1\min \\
 & \Rightarrow \dfrac{1}{{{10}^{5}}}km=1cm \\
\end{align}$
So, we have speed as,
$\begin{align}
  & =\dfrac{\dfrac{315\times \pi \times 90cm}{{{10}^{5}}}}{\dfrac{1}{60}}\dfrac{km}{hr} \\
 & =\dfrac{315\times \pi \times 90\times 60}{{{10}^{5}}}\dfrac{km}{hr} \\
\end{align}$
Now, we will substitute $\pi =\dfrac{22}{7}$. So, we have,
$\begin{align}
  & speed=\dfrac{315\times 22\times 90\times 60}{7\times {{10}^{5}}}km/hr \\
 & =\dfrac{45\times 22\times 90\times 60}{{{10}^{5}}}km/hr \\
 & =\dfrac{5346000}{{{10}^{5}}}km/hr \\
 & =53.46km/hr \\
\end{align}$
So, the speed of the bus is 53.46 km/hr.

Note: To solve these types of questions one has to have a basic understanding of time, distance and speed. Also, it is important to note that the distance covered by a wheel in 1 complete rotation is equal to its circumference and also students should know the definitions and formulas related to the circle.