| Assertion (A): Circular footpath of width \[2\,m\] is constructed at the rate of \[\text{Rs}.\text{2}0\text{ per }{{\text{m}}^{2}}\]around a circular park of radius\[\text{15}00\text{ m}\]. The total cost of construction of the footpath is \[\text{Rs}.\text{377}0\text{51}.\text{2}\]. |
| Reason (R): Area of the footpath \[=\pi ({{R}^{2}}-{{r}^{2}}),\] where R and rare outer and inner radii of the park respectively. |
A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
C) Assertion (A) is true but Reason (R) is false
D) Assertion (A) is false but Reason (R) is true
Correct Answer: A
Solution :
| [a] Radius of the park (inner radius), |
| \[\text{r}=\text{15}00\text{ m}\] |
| Width of the footpath around the park \[=\text{2 m}\] |
| Let R be the outer radius of the park including the footpath. |
| Then, \[R=(1500+2)m=1502m\] |
| Now, area of the footpath \[=\pi {{r}^{2}}-\pi {{r}^{2}}\] |
| \[=\pi \{{{R}^{2}}-{{r}^{2}}\}=3.14\{{{(1502)}^{2}}-{{(1500)}^{2}}\}\] |
| \[=3.14\{(1502+1500)(1502-1500)\}\] |
| \[=3.14\times 6004=18852.56c{{m}^{2}}\] |
| \[\therefore \] Total cost of construction of the footpath at the rate of |
| \[Rs.20per\,{{m}^{2}}=20\times 18852.56=Rs.377051.2\] |
| \[\therefore \] Assertion: True; Reason: True and it is the correct explanation of Assertion. |
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