A square sheet of cardboard 100 cm by 100 cm is to have corners of side length x cut out. These cutouts will create flaps that can be folded up to form a box with no lid. The length and width of the base of the box are given by the expression (100 – 2x), and the height of the box is x, where x is in centimetres.

a) Express the volume of the box as a function of x. (2 marks)
b) Express the surface area of the box as a function of x. (2 marks)
c) What is a simplified expression for the ratio of the volume of the box to its surface area? Identify any non-permissible values of x. (2 marks)

My work for a,b,c:

a)(100-2x)x would be function of volume with a x.

b)I'm not sure about this one, I need help here

c) Same here because I'm not sure and I need help

If anyone could please help me and show me where I went wrong with this, I will be extremely grateful for your help with my math!

1 answer

(a) You need 3 dimensions for volume. What you have is the area of one of the sides of the box. You need

(100-2x)(100-2x)x = x(100-2x)^2

(b) You need the area of the base, plus the area of the 4 sides:

(100-2x)^2 + 4x(100-2x) = (100-2x)(100+2x)

(c) Now that you have the two expressions, the ratio is

x(100-2x)^2 / (100-2x)(100+2x) = x/(100+2x)