Cube to cuboid

Geometry Level 3

The golden cube (left photo) has a side length of $6$ . It is to be melted to form a golden cuboid (right photo) with a base dimension of $18 \times 6$ . What is the ratio of the surface area of the cube to the surface of the cuboid? If your answer is of the form $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

Note:

Assume no material is wasted.

The answer is 22.

1 solution

A Former Brilliant Member
Jun 17, 2017

Volume of the cube:

$V_{cube}=6^3=216$

Surface area of the cube:

$S_{cube}=6(6^2)=6(36)=216$

The cube and cuboid must have equal volumes:

$V_{cuboid}=18(6)h$

$216=108h$

$h=2$

Therefore, the dimension of the cuboid is $18 \times 6 \times 2$ . And surface area is

$S_{cube}=2[2(18)+2(6)+18(6)]=2(160)=312$

The ratio of the surface areas is,

$ratio=\dfrac{216}{312}=\dfrac{9}{13}$

Finally,

$\large a+b=9+13=$ $\large \boxed{22}$

Cube to cuboid

The answer is 22.

1 solution

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