An open-top box with a square base is to be constructed to hold a fixed volume

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

An open-top box with a square base is to be constructed to hold a fixed volume VV. The material for the base costs three times as much per square unit as the material for the sides. If the base edge length is xx and the height is hh, which of the following ratios of height to base edge minimizes the total cost of materials? (a) hx=23\dfrac{h}{x}=\dfrac{2}{3} (b) hx=32\dfrac{h}{x}=\dfrac{3}{2} (c) hx=12\dfrac{h}{x}=\dfrac{1}{2} (d) hx=3\dfrac{h}{x}=3

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