An open‐top box with a square base is to be constructed so that its volume is 3,200 cm

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

Problem

An open‐top box with a square base is to be constructed so that its volume is 3,200 cm3^{3}. After expressing the surface area in terms of the base edge length xx (in cm), the design team obtained the single‐variable function S(x)=2x2+640xfor x>0.S(x)=2x^{2}+\frac{640}{x} \quad\text{for } x>0. Here S(x)S(x) is the exterior surface area in cm2^{2}. (a) Determine the value of xx that yields the minimum surface area. Show all work, including any calculus used. (b) Calculate the minimum surface area, rounding your answer to the nearest whole square centimeter. (c) Explain why the value of xx you found indeed produces a minimum rather than a maximum for the surface area within the domain.

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