73
A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below. The radius of circular piece of land is 35 m. Based on given information, answer the following questions: (i) Find the length of wire needed to fence the entire land. (ii) Find the length of each side of the square land on which house will be constructed. (iii) (a) The farmer wishes to grow grass on the shaded region around the house. Find the cost of growing the grass at the rate of ₹ 50 per square metre. OR (iii) (b) Find the ratio of area of land on which house is built to remaining area of circular piece of land.
Show SolutionHide Solution↓
(i) Length of wire $= 2 \times \frac{22}{7} \times 35 = 220$ m
(ii) Diagonal of square $= 70$ m. Length of each side of the square land $= \frac{70}{\sqrt{2}}$ or $35\sqrt{2}$ m
(iii) (a) Area on which grass is grown = Area of two segments $= 2 \times [\frac{90}{360} \times \frac{22}{7} \times 35 \times 35 - \frac{1}{2} \times 35 \times 35] = 700$ m$^2$. Cost of growing the grass $= 700 \times 50 = \text{Rs} 35000$
(iii) (b) Required ratio $= \frac{\text{area of square}}{\text{area of circle} - \text{area of square}} = \frac{35\sqrt{2} \times 35\sqrt{2}}{\frac{22}{7} \times 35 \times 35 - 35\sqrt{2} \times 35\sqrt{2}} = \frac{2450}{1400}$ or $\frac{7}{4}$. $\therefore$ Required ratio is $7:4$
(ii) Diagonal of square $= 70$ m. Length of each side of the square land $= \frac{70}{\sqrt{2}}$ or $35\sqrt{2}$ m
(iii) (a) Area on which grass is grown = Area of two segments $= 2 \times [\frac{90}{360} \times \frac{22}{7} \times 35 \times 35 - \frac{1}{2} \times 35 \times 35] = 700$ m$^2$. Cost of growing the grass $= 700 \times 50 = \text{Rs} 35000$
(iii) (b) Required ratio $= \frac{\text{area of square}}{\text{area of circle} - \text{area of square}} = \frac{35\sqrt{2} \times 35\sqrt{2}}{\frac{22}{7} \times 35 \times 35 - 35\sqrt{2} \times 35\sqrt{2}} = \frac{2450}{1400}$ or $\frac{7}{4}$. $\therefore$ Required ratio is $7:4$