Given a 10 cm by 20 cm piece of paper, if you cut out four equal squares from the corners and fold up the resulting sides you will create an open box. Write an expression for the volume of this box in terms of the side length of each cut‐out square, x.
original: 10 cm by 20 cm
after cutting x cm at the corners
new length = 20 - 2x
new width = 10 - 2x
height = x
volume = x(10-2x)(20-2x) , 0 < x < 10
To find the volume of the box, we need to consider its dimensions. Since equal squares are cut out from each corner, the resulting dimensions of the box will be:
Length: 20 cm - 2x (each cut-out reduces the length by 2x)
Width: 10 cm - 2x (each cut-out reduces the width by 2x)
Height: x cm (since the cut-out squares are folded to form the height)
So the expression for the volume of the box can be calculated by multiplying the length, width, and height:
Volume = Length * Width * Height
= (20 cm - 2x) * (10 cm - 2x) * x
= (200 cm^2 - 40x cm - 20x cm + 4x^2 cm^2) * x
= (4x^2 - 60x + 200) * x
= 4x^3 - 60x^2 + 200x
Thus, the expression for the volume of the box in terms of the side length of each cut-out square, x, is 4x^3 - 60x^2 + 200x.