A rectangular box with a square base and no restriction on its height is to have a…

MATHEMATICS · Subjective · A Level · Mathematics · Differentiation · Differentiation

Problem

A rectangular box with a square base and no restriction on its height is to have a volume of

32000\,\text{cm}^3

. The material for the base costs Rs

3

per cm

^{2}

, the material for the four vertical sides costs Rs

1

per cm

^{2}

, and the material for the lid costs Rs

2

per cm

^{2}

. (i) Denoting the side of the square base by

x\,\text{cm}

and the height by

h\,\text{cm}

, express the total cost

C(x)

of constructing the box in terms of

x

alone. (ii) Find the value of

x

that minimises the total cost. Hence obtain the corresponding height

h

. (iii) Justify that the critical point obtained in part (ii) yields the absolute minimum cost. (iv) Compute the minimum total cost of the box.