Multiple choice

A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm is melted and recast into 8 cubes of the same volume. What is the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes?

  1. 4 : 7

  2. 7 : 8

  3. 7 : 12

  4. 2 : 3
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B Correct answer
Explanation

Volume of cuboid = 18 * 36 * 72 = 46656. Volume of each cube = 46656 / 8 = 5832. Side of cube = cube root of 5832 = 18. Surface area of cuboid = 2(18*36 + 36*72 + 72*18) = 2(648 + 2592 + 1296) = 2(4536) = 9072. Lateral surface area of one cube = 4 * side^2 = 4 * 18^2 = 4 * 324 = 1296. Total lateral area of 8 cubes = 8 * 1296 = 10368. Ratio = 9072 / 10368 = 7/8.

AI explanation

The total surface area of the cuboid is calculated as 2 times (18 times 36 plus 36 times 72 plus 72 times 18), giving 9072 square cm. The volume of the cuboid is 46656 cubic cm, meaning each of the 8 cubes has a volume of 5832 cubic cm and an edge length of 18 cm. The lateral surface area of one such cube is 4 times 18 squared, and multiplying by 8 cubes gives a total lateral surface area of 10368 square cm. The ratio of the total surface area of the cuboid to the total lateral surface area of the 8 cubes is 9072 to 10368, which simplifies to 7 to 8.

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