Question

Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that ∠PAB = 30° as shown in the following figure, where O is the centre of the semicircle. In part I, he planted saplings of the Mango tree, in part II, he grew tomatoes, and in part III, he grew oranges.

Based on the given information, answer the following questions.

1. What is the measure of ∠POA?   (1)
2. Find the length of wire needed to fence the entire piece of land.   (1)
3. 1.  Find the area of the region in which saplings of the Mango tree are planted.   (2)
                                       OR
   2. Find the length of wire needed to fence the region III.  (2)

Answer 1

Diameter = 70 m

∠PAB = 30°

i. In ΔPОА,

OA = OP = r

∠PAO = ∠APO = 30°

∠POA + ∠PAO + ∠APO = 180°

∠POA + 30° + 30° = 180°

∠POA = 120°

ii. Length of wire = d + πr

= `70 + 22/7 xx 35`

= 70 + 110

= 180 m

iii. a.

∠1 = 180° – 120 = 60°

∠2 = ∠3 = 60°   .....(OP = OB)

∠1 + ∠2 + ∠3 = 180°

60° + 2∠2 = 180°

2∠2 = 120°

∠2 = 60° = ∠3

ar (Mango) = ar (I)

= ar (segment)

= `θ/360 πr^2 - 1/2 r^2 sin θ`    ......(∵ θ = 60)

= `60/360 xx 22/7 xx 35 xx 35 - 1/2 xx 35 xx 35 sin 60°`

= `1225(1/6 xx 22/7 - sqrt3/4)`

= `1225/2(22/21 - sqrt3/2)`

= 612.5 (1.05 – 0.865)

= 612.5 × 0.185

= 113.31 m²

OR

b. Length of wire region III = Length of arc

= `θ/360 xx 2πr + AP`     .....(∵ θ = 120°)

= `(120°)/(360°) xx 2 xx 22/7 xx 35 + 35sqrt3`

= `(220°)/3 + 35 xx 1.73`

= 73.33 + 60.55

= 133.8 m