question_answer

There is a circular plot of radius 7 metres. A circular, path surrounding the plot is being gravelled at a total cost of Rs. 1848 at the rate of Rs. 4 per square metre. What is the width of the path? (in metres)

A)  7

B)  11

C)  9

D)  21

E)  14

Correct Answer: A

Solution :

      Total expenses on path construction = Rs. 1848 Rate = Rs. 4 per sq. metre \[\therefore \]Area of path \[=\frac{1848}{4}\]= 462 sq. metre If the width of path be r metre, then                         \[\pi {{(7+r)}^{2}}-\pi \times {{7}^{2}}=462\] \[\Rightarrow \]\[\frac{22}{7}({{(7+r)}^{2}}-{{7}^{2}})=462\] \[\Rightarrow \]\[(49+{{r}^{2}}+14r-49)=\frac{462\times 7}{22}\] \[\Rightarrow \]\[{{r}^{2}}+14r=147\] \[\Rightarrow \]\[{{r}^{2}}+14r=147=0\] \[\Rightarrow \]\[{{r}^{2}}+21r-7r-147=0\] \[\Rightarrow \]\[r(r+21)-7(r+21)=0\] \[\Rightarrow \]\[(r-7)(r+21)=0\] \[\Rightarrow \]\[r=7\,\text{because}\,\text{r}\ne -21\]