The combined volume of two cubes of different sizes, each having edges of integer values, is equal to the combined length of all their edges. What are the dimensions of the two cubes?
Each cube has 12 congruent edges; call them a and b.
So the volume of the cube on the left + the volume of the cube on the right = the sum of the lengths of the edges on the left cube + the sum of the lengths of the edges on the right cube.
That is, a3 + b3 = 12a + 12b.
(a + b)(a2 – ab + b2) = 12(a + b)
a2 – ab + b2 = 12
a2 – ba + (b2 -12) = 0
Using the quadratic formula, we have:
So the volume of the cube on the left + the volume of the cube on the right = the sum of the lengths of the edges on the left cube + the sum of the lengths of the edges on the right cube.
That is, a3 + b3 = 12a + 12b.
(a + b)(a2 – ab + b2) = 12(a + b)
a2 – ab + b2 = 12
a2 – ba + (b2 -12) = 0
Using the quadratic formula, we have:
| a = | b ± √b2 – 4(b2 – 12) | = | b ± √b2 – 4b2 + 48 | = | b ± √48 – 3b2 |
| 2 | 2 | 2 |
But recall, the edges are positive integers. We try values for b and find b = 2 or 4.
| b = 2 → a = | 2 ± √48 – 12 | = | 2 ± 6 | = 4 (we take the positive value) |
| 2 | 2 |
| b = 4 → a = | 4 ± 0 | = 2. |
| 2 |
So a and b are both either 2 or 4. However, b > a so b = 4 and a = 2.