Reese Tran
09/08/2024 · High School
Use Lagrange Multipliers to find the volume of the largest open-top box that has \( 120 \mathrm{~m}^{\wedge} 2 \) of surface area, and 70 m of edge length (sum of all edges). Recall that for two constraint functions \( \mathrm{g} 1, \mathrm{~g} 2 \), the method of Lagrange multipliers states that \( \nabla \mathrm{f}=\lambda 1 \nabla \mathrm{~g} 1+\lambda 2 \nabla \mathrm{~g} 2 \).
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The maximum volume of the open-top box is approximately \( 60.84 \, \text{m}^3 \) when \( x \approx 7.8 \, \text{m} \) and \( h \approx 1.9 \, \text{m} \).
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