Mathematics

Problem

A square sheet of metal of side

20\,\text{cm}

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

x\,\text{cm}

from each corner and folding up the sides. a. (i) Show that the volume

V(x)

of the box can be written as

V(x)=4x(10-x)^{2}

. b. (ii) State the interval of

x

values that need to be considered for this problem. c. (iii) Find the value of

x

that gives the maximum volume of the box. d. (iv) Calculate the maximum volume (in cubic centimetres).