A box with a square base and no top is to be constructed

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A box with a square base and no top is to be constructed. Exactly

48\text{ m}^2

of material is available for the four sides and the base. Let

s

be the length of one side of the square base (in meters), let

h

be the height of the box (in meters), and let

V(s)

be the volume (in cubic meters) written solely as a function of

s

for

s>0

. (a) Derive an expression for

V(s)

in terms of

s

using the surface–area constraint. State the permissible domain of

s

. (b) Determine the value of

s

that maximizes

V

and calculate the corresponding height

h

. Show complete calculus justification. (c) Verify that the critical point obtained in part (b) yields a maximum by applying a second–derivative test or an equivalent argument based on concavity. (d) Provide the exact maximum volume of the box, simplified and expressed with correct units.

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