UPSC CDS 1 2022 Math Paper

Given:
The ratio of the cuboid length, breadth, and height is 27:8:1, respectively.
Total Surface area of cuboid =p
Total surface area of cube =q
Formula Used:
Volume of cuboid = length × breadth × height
Volume of cube =(‌ side ‌)3
Total surface area of cuboid =2(lb+bh+hl)
Total surface area of cube =6(‌ Side ‌)2
Calculation:
Let the cuboid length, breadth, and height be 27k,8k,k respectively.
The total surface area of cuboid =2(27k×8k+8k×k+k×27k)
⇒p=2k2(216+8+27)
⇒p=2k2(251)
⇒p=502k2
The cuboid is melted and recast into a cube of side 'a'. ⇒ Volume of cuboid = Volume of a cube
⇒1×b×h=a×a×a
⇒27k×8k×k=a3
⇒216k3=a3
⇒a=6k
Now the total surface area of a cube
q=6(6k)2
⇒q=216k2
Therefore, the required ratio
‌

p

q

=‌

502k2

216k2

⇒‌

p

q

=‌

251

108

∴ The required value of ‌

p

q

is ‌

251

108