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

Hint:

First find the actual dimensions of the cuboid using the sum-of-edges condition, then compare surface areas.

Answer 1

The Correct Option is C

Step 1: Find the dimensions of the cuboid. A cuboid has 12 edges, 4 of each length l, b, h, so \(4(l+b+h) = 208\), giving \(l+b+h = 52\).
Step 2: Use the given ratio. The sides are in the ratio 3 : 4 : 6, so let the sides be \(3k, 4k, 6k\). Then \(13k = 52\), so \(k = 4\). The dimensions are \(l = 12\) cm, \(b = 16\) cm, \(h = 24\) cm.
Step 3: Find the volume of the cuboid. Volume \(= 12 \times 16 \times 24 = 4608\) cm\(^3\).
Step 4: Find the number of small cubes. Each small cube has volume \(2^3 = 8\) cm\(^3\), so the number of cubes formed is \(4608 / 8 = 576\).
Step 5: Find the surface area of the cuboid. Surface area \(= 2(lb+bh+hl) = 2(12\times16 + 16\times24 + 24\times12) = 2(192+384+288) = 2 \times 864 = 1728\) cm\(^2\).
Step 6: Find the total surface area of the small cubes. Each small cube has surface area \(6 \times 2^2 = 24\) cm\(^2\), so the total is \(576 \times 24 = 13824\) cm\(^2\).
Step 7: Find the ratio. \(1728 : 13824\). Dividing both by 1728 gives \(1 : 8\).\[\boxed{1:8}\]