Question

Jack cuts four squares with side length X inches from corners by a 5" x 7" cardboard rectangle. He folds the remaining cardboard to make a tray that is x inches high. Write and simplify a function for the volume V of the tray in terms of X.

Answer 1

`V = x^3-24x^2+35x`

Explanation:

Jack makes a cuboid. If the length is l, the width is w and the height is h, then its volume is

`V = lwh`

The height is determined by the side of the square cut out of each corner.

`h = x`

Along the length of the width, the length of two sides of the square are cut.

`l = 7 - 2x`

`w = 5 - 2x`

`V = lwh = (7-2x)(5-2x)(x)`

`V = x(35-24x+4x^2) = x(x^2-24x+35) = x^3-24x^2+35x`

Answer 2

Explanation:

Given:

Dimensions of rectangle cardboard:

Length, l = 7 inches

Width, b = 5 inches

if the square of sides x inches is cut from each corner of the rectangular cardboard and folded to make a tray, therefore:

Length of the tray = length of cardboard - 2 (side of the square )

= 7 - (2 × x)

= 7 - 2x inches

Width of the tray = width of cardboard - 2(side of square)

= 5 - (2 × x)

= 5 - 2x inches

height of tray = cut sides of square

= x inches

volume of tray = Length × width × height

= (7 - 2x) × (5 - 2x) × x

Volume of tray = 35x - 10x^2 - 14x^2 + 4x^3

Simplifying further,

= (35x - 24x^2 + 4x^3) cubic inches