Question

A man by walking diametrically across a circular grass plot takes 45 seconds less than if he had kept to the circular path round the outside. He walks 80 metre a minute. What is the diameter of the grass plot. ($\mathrm{\pi }-\frac{22}{7}$)

Answer 1

Let the radius of the circle be r m.
Let the time taken by the man to walk around the circular path be t s.
Given speed = $80\mathrm{m}/\min =\frac{80}{60}\mathrm{m}/s=\frac{4}{3}\mathrm{m}/\mathrm{s}$


$$\begin{gathered} \mathrm{So},\mathrm{speed}\mathrm{while}\mathrm{covering}\mathrm{the}\mathrm{diametrical}\mathrm{distance}=\frac{\mathrm{distance}}{\mathrm{time}} \\ =\frac{2r}{t-45}(\because \mathrm{it}\mathrm{takes}45\mathrm{secs}\mathrm{less}\mathrm{than}\mathrm{the}\mathrm{circular}\mathrm{path}) \\ =\frac{4}{3}\mathrm{m}/\mathrm{s}...\left(1\right) \end{gathered}$$

Now, speed while covering the circular distance = $\frac{\mathrm{distance}}{\mathrm{time}}=\frac{\mathrm{\pi }r}{t}=\frac{4}{3}\mathrm{m}/\mathrm{s}...\left(2\right)$

Now, we have:

$\frac{\mathit{\pi }\mathit{r}}{\mathit{t}}=\frac{2r}{(t-45)}=\frac{4}{3}[\mathrm{From}(1)\mathrm{and}(2\left)\right]$
$$\begin{gathered} \Rightarrow \pi (t-45)=2t \\ \Rightarrow \pi t\mathit{}-45\pi =2t \\ \Rightarrow (\frac{22}{7}-2)t\mathit{}=45\times \frac{22}{7} \\ \Rightarrow 8t=45\times 22 \\ \Rightarrow t=\frac{45\times 22}{8}=\frac{495}{4} \end{gathered}$$

$$\begin{gathered} \mathrm{So},\frac{\pi r}{t}=\frac{4}{3} \\ \Rightarrow r=\frac{4}{3}\times \frac{t}{\pi } \\ \Rightarrow r=\frac{495\times 7}{22\times 3}=52.5 \\ \therefore 2r=52.5\times 2=105\mathrm{m} \end{gathered}$$

Hence, the diameter of the grass plot is 105 m.