Problem Solving

vinay1983 wrote:A large cube is formed from the material obtained by melting 3 cubes of sides 3 units, 4 units and 5 units. What is the ration of the sum of the total surface area of the small cubes to that of the large cube?

A. 1:1
B. 12:7
C. 25:18
D. 12:13
E. 12:17

Total volume of all 3 cubes
(3x3x3) + (4x4x4) + (5x5x5) = 27 + 64 + 125 = 216
So, the NEW cube must have volume of 216 cubic units
If we let k be length of one side of NEW cube, then we know that k^3 = 216
Solve, to get k = 6
So, the NEW cube has dimensions 6x6x6

Total surface of the 3 original cubes
6(3x3) + 6(4x4) + 6(5x5) = 6(9) + 6(16) + 6(25)
= 6(50)

Total surface of the NEW cube
6(6x6) = 6(36)

The ratio of total surface area of the small cubes to that of the NEW cube = 6(50)/6(36)
= 50/36
= [spoiler]25/18[/spoiler]
= C

Cheers,
Brent

shefdsouza wrote:Why do you do 6(old cube surface)+6(old cube surface)+^(old cube surface)?
Why is it 6?? When 6 is the side of the new big cube?

A cube has 6 faces, so if the length of one edge of a cube is k, each of the six faces has area k^2, and the total surface area of the cube is 6k^2.

So the number '6' appears in the solution to this problem for two completely different reasons -- you'll always multiply by 6 when you calculate the surface area of any cube, but in this question '6' coincidentally also happens to be the length of an edge of the largest cube.

Matt@VeritasPrep wrote:I'm not sure about the stem here: it's not clear whether the cube is made by gluing the cubes together in some way (like a tower of cubes) or by breaking in down into 216 unit cubes, then using those to form a giant cube. I guess the intention is the second ... but even then 'melting' gives such a strange (and wrong) impression of what's happening here.

If we picture the cubes being made out of some metal or plastic that can be melted into liquid state, then poured into a new mold without changing total volume, this could make sense.