## An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. What dimensions minimize the amount of material requi

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Answer:x = 8 fth = 1/2 ftStep-by-step explanation:Let x be side of the base then area of the base is x²Let h be the height of the tankTank volume is 32 ft³ and is 32 = x²*h then h = 32 /x²Area of base + lateral area = total area (A)A = x² + 4*x*h ⇒ A = x² + 4*x*(32/x²) A = x² + 128/xA(x) =

x² + 128/x (1)Taking derivatives on both sides of the equationA´(x) = 2x – 128/x² A´(x) = 0 2x – 128/x² = 0(2x² -128) / x² = 02x² – 128 = 0x² =√64x = 8 ftThe result is minimum since replacing in equation (1) x = 8 we getA(x) > 0Andh = 32/x²h = 1/2 ft