Question

A solid metallic cuboid with sides in the ratio 3 : 4 : 6 is melted to form smaller cubes with sides 2 cm. If the sum of the length of the edges of the cuboid is 208 cm, then what is the ratio of the surface area of the original cuboid to the total surface area of the smaller cubes?

• A. 1 : 6
• B. 4 : 11
• C. 1 : 8
• D. 2 : 9
• E. 3 : 11

Hint:

Use the edge-sum to find the cuboid's actual dimensions, then compare its total surface area to the combined surface area of all the small cubes it is melted into.

Answer 1

The Correct Option is C

Step 1: Find the dimensions of the cuboid.
A cuboid has 12 edges: 4 of each dimension, so the sum of all edges \(=4(l+w+h)\).
\(4(l+w+h)=208 \Rightarrow l+w+h=52\)
Given ratio \(3:4:6\), let \(l=3k, w=4k, h=6k\).
\(3k+4k+6k=52 \Rightarrow 13k=52 \Rightarrow k=4\)
So \(l=12\) cm, \(w=16\) cm, \(h=24\) cm.

Step 2: Find the number of smaller cubes.
Volume of cuboid \(=12\times16\times24=4608\) cm\(^3\)
Volume of one small cube (side 2 cm) \(=2^3=8\) cm\(^3\)
Number of small cubes \(=4608/8=576\)

Step 3: Find the surface area of the original cuboid.
Surface area \(=2(lw+wh+hl)=2(12\times16+16\times24+24\times12)\)
\(=2(192+384+288)=2(864)=1728\) cm\(^2\)

Step 4: Find the total surface area of all the smaller cubes.
Surface area of one small cube \(=6\times2^2=24\) cm\(^2\)
Total surface area of 576 cubes \(=576\times24=13824\) cm\(^2\)

Step 5: Form the ratio.
Ratio \(=1728:13824\). Dividing both by 1728: \(1:8\)

So the required ratio is 1 : 8, matching option (c).