Hello, ASSC!

There is a verysneakyway to solve this one . . .

The combined heights of two cubes are 11 metres

and their combined volumes equal 407 cubic metres.

Calculate the dimensions of each cube.

Letxandybe the sides of the two cubes.

Then we have:

. . (1) . .x + y . = . 11

. . (2) .x³ + y³ .= .407

Cube (1): . . . . . . (x + y)³ .= .11³

. . . x³ + 3x²y + 3xy² + y³ .= .1331

. . . (x³ + y³) + 3xy(x + y) .= .1331

. . . . . .↓ . . . . . . . . . ↓

. . . . (407). . + . 3xy(11) . .= .1331

This simplifies to: .xy .= .28 . → . y .= .28/x

Substitute into (1): . x + 28/x .= .11

. . and we have: . x² - 11x + 28 .= .0

. . which factors: . (x - 4)(x - 7) .= .0

. . and has roots: . x = 4, 7 . → . y = 7, 4

Therefore, the cubes have sides 4 metres and 7 metres.