Mathematics

Problem

A square sheet of cardboard measuring

40\text{ cm}

on each side is to be converted into an open-top box by cutting out equal squares of side length

x

cm from each corner and folding up the resulting flaps. (a) Express the volume

V

of the box as a function of

x

. (b) Determine all critical values of

V(x)

that are feasible for the physical box. (c) Identify the value of

x

that yields the maximum volume and calculate this maximum volume. Give both answers correct to two decimal places. a. Part (a) -- model

V(x)

b. Part (b) -- locate critical values c. Part (c) -- select and evaluate the maximum