The combined hieghts of two cubes are 11 metres and there combined volumes equal 407 cubic metres calculate the dimensions of each cube. need working out pleasee.
Hello, ASSC!
There is a very sneaky way 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.
Let x and y be 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.