Q) 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) 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 meter.
(iv) Find the ratio of area of land on which house is built to remaining area of circular piece of land.
PYQ: 36 – CBSE 2025 – Code 30 – Series 5 – Set 1
Ans:
Step 1: Let’s make a simpler diagram for our better understanding of the question:
(i) Find the length of wire:
It is given that the land is circular in shape with radius of 35 m
Since fence is required to be done around the entire land, it will go along the circular periphery.550 m
Hence, length of the wire = circumference of the circular land
= 2 π R = 2 x (22 / 7) x 35
= 2 x 22 x 5 = 220 m
Therefore, the length of the fire required to fence the entire land is 220 m.
(ii) Find the length of each side of the square land on which house will be constructed.
Let’s understand that largest possible square will be drawn inside a circle when its diagonal is equal to the diameter of the circle.
∵ Diameter of the circle = 2 x Radius of the circle
= 2 x 35 = 70 m
∴ Diagonal of the largest square = 70 m
Next, let’s calculate the area of the square:
We know that, when a square has side as A, then its diagonal is A√2
Here, Diagonal is 70 m, hence, A √2 = 70
∴ A = 70 / √2 = 35 √2
Therefore, the side of the largest possible square inside the circular land is 35√2 m.
(iii) Cost of growing the grass on the shaded region around the house:
Let’s first calculate the area around the house and then calculate area of the shaded region:
Area of the entire region around the house = [Area of circle – Area of the square]
= [Area of circle – Area of the square]
= [π R2 – A2]
= [(22/7) (35)2 – (35 √2)2] = (35)2 [(22/7) – (√2)2]
= (35)2 [(22 / 7) – 2] = (35)2 (8 / 7)
= (35 x 35 x 8) / (7)
= 5 x 35 x 8 = 1400 m2
Now this area around the house is equally divided in 4 regions around the house and 2 of them are being used for growing grass (shown as shaded regions in the question’s diagram)
Therefore, Area of shaded region for growing frass = (1/2) x 1400 = 700 m2
Next, the cost of growing the grass in the 2 shaded regions = Area of the region x cost of grass per m2
∵ The grass cost is at the rate of Rs. 50 per square meter
∴ The cost of growing the grass = 700 x 50
= Rs. 35,000
Therefore, the cost of growing grass in the shaded regions is Rs. 35,000.
(iv) Ratio of area of house land to remaining area of circular piece of land:
Area of land of house = Area of the square = A2
Since A = 35√2 m (calculated in part (ii) above)
∴ Area of land of house = (35√2 )2 = 1225 x 2 = 2450 m2
Remaining area of circular piece of land = 1400 m2 (calculated in part (iii) above)
∴ Ratio = (Area of house land / Remaining area of circular piece of land)
= 2450 / 1400 = 35 x 35 x 2 / 35 x 5 x 8 = 7 / 4
Therefore, the Ratio of area of house land to remaining area of circular piece of land is 7 : 4.
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