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What is the maximum volume in cubic inches of an open box to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides?

Respuesta :

let x be length of side of each cut -out square

V =  x(16 - 2x)(30 - 2x)  =  x(480 - 92x + 4x^2) = 4x^3 - 92x^2 + 480x

dV/dx = 12x^2 - 184x + 480  = 0 for turning points

x =  12, 3 .333...

second derivative = 24x - 184

when x = 12 this is positive and when x = 3.33... its negative so we have a maximum when x = 3.33...

Maximum volume  is  3.3333..( 16 - 2(3.33...)(30 - 3.33....)

 =   725.92 in^3

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