Question

11. The volume, in cubic inches, of a rectangular box can be expressed as the product

of its three dimensions and with the function V(x) = x3 - 16x2 + 79x - 120. The
length of the box is represented by the expression x - 8. Find linear expressions
with integer coefficients for the width and height. Hint: The width is greater
than the height.

Answer 1

The answer is:
`w = x - 3, h = x - 5`

`V = l * w * h`

`V = x^3 - 16x^2 + 79x - 120`

`l = x - 8`

`x^3 - 16x^2 + 79x - 120 = (x - 8) * w * h`

`w * h = ((x^3 - 16x^2 + 79x - 120) )/((x - 8))`

`x^3 - 16x^2 + 79x - 120`

`x - 8 (* x^2)\ | \ x^3 - 8x^2`

_________________________ (subtract)

`-8x^2 + 79x - 120`

`x - 8 (* -8x) \ | -8x^2 + 64x`

_________________________ (subtract)

`15x - 120`

`x - 8 (* 15) \ | \ 15x - 120`

_________________________ (subtract)

`0`

`w * h = ((x^3 - 16x^2 + 79x - 120))/((x - 8) = x^2 - 8x + 15)`

Let's find factors of
`x^2 - 8x - 15`:

`x^2 - 8x - 15 = x^2 - 3x - 5x + 15`

`= x * x - 3 * x - (5 * x - 5 * 3)`

`= x(x - 3) - 5(x - 3)`

`= (x - 3)(x - 5)`

`w * h = (x - 3) * (x - 5)`

`w > h` and
`(x - 3) > (x - 5)`

`\longrightarrow w = x - 3`

`h = x - 5`