question_answer

Semi-circular lawns are attached to the edges of a rectangular field measuring 42 m \[\mathbf{\times }\] 35 m. Find the area of the total field.

A) \[3895.6\text{ }{{m}^{2}}\]                

B) \[3818.5\text{ }{{m}^{2}}\]

C) \[3735.6\text{ }{{m}^{2}}\]                

D) \[3899.9\text{ }{{m}^{2}}\]

E) None of these

Correct Answer: B

Solution :

Explanation Area of lawn = Area of rectangle ABCD + 2 x Area of semicircle with diametre AB + 2 x Area of semi-circle with diametre \[\operatorname{BC} = Length \times  breadth +2\left( \frac{\pi {{R}^{2}}}{2} \right)+2\,\left( \frac{\pi {{r}^{2}}}{2} \right)\] Where R = radius of bigger semicircle = \[=\,\,\frac{42}{2}=21\,m\] \[\operatorname{r} = radius of smaller semicircle =\,\,\frac{35}{2}\,m\] Now, area of lawn \[=\,\,42\times 35+\frac{22}{7}\times 21\times 21+\frac{22}{7}\times \frac{35}{2}\times \frac{35}{2}\] \[= 1470 + 1386 + 962.5 = 3818.5 {{m}^{2}}.\]