UPSC CDS I 2025 Mathematics Paper

Inner radius of the hollow sphere (R)=20cm
Height of the pot (h)=30cm
From the previous calculation, the radius of the circular opening (r)=10√‌3cm
From the previous calculation, the perpendicular distance from the center of the sphere to the plane of the circular opening (d)=10cm
Calculations:
The perpendicular distance from the center of the sphere to the plane of the circular opening (d) as the adjacent side to the angle θ (this is the vertical line segment from the center to the center of the opening).
The inner radius of the circular opening ( r ) as the opposite side to the angle θ.
We can use trigonometric ratios:
‌cos(θ)=‌ Adjacent ‌∕‌ Hypotenuse ‌=d∕R
‌cos(θ)=d∕R=10cm∕20cm=1∕2
The angle whose cosine is 1∕2 is 60∘.
⇒θ=60∘=π∕3