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Mathematics · 2025 · 4 marks
CBSE 2025 · Region 6 · Set 1 · Q36
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…
Anurag purchased a farmhouse which is in the form of a semicircle of diameter $\displaystyle 70$ m. He divides it into three parts by taking a point P on the semicircle in such a way that $\displaystyle \angle \mathrm{PAB}=30^{\circ}$ as shown in the following figure, where O is the centre of semicircle.
In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.
(i)
What is the measure of ∠POA ?
(ii)
Find the length of wire needed to fence entire piece of land.
(iii)
(a)
Find the area of region in which saplings of Mango tree are planted.
OR
(b)
Find the length of wire needed to fence the region III.
Marking-scheme solution
(i) \(\displaystyle \angle \mathrm{POA}=120^{\circ}\)
(ii) Length of wire needed to fence entire piece of land \(\displaystyle =\frac{22}{7} \times 35+70=180 \mathrm{~m}\)
(iii) Required area \(\displaystyle =\frac{60}{360} \times \frac{22}{7} \times(35)^{2}-\frac{\sqrt{3}}{4} \times(35)^{2}\)
\[=\left(\frac{1925}{3}-\frac{1225 \sqrt{3}}{4}\right) \mathrm{m}^{2} \text { or } 111.89 \mathrm{~m}^{2} \text { (approx.) }
\]
(iii) In \(\displaystyle \triangle \mathrm{APB}, \frac{\mathrm{AP}}{\mathrm{AB}}=\cos 30^{\circ}\)
\[\mathrm{AP}=35 \sqrt{3} \mathrm{~m}
\]
\[=\left(\frac{220}{3}+35 \sqrt{3}\right) \mathrm{m} \text { or } 133.8 \mathrm{~m} \text { (approx.) }
\]
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