reza is helping en shah to make a box without the top. the box is made by cutting away four squares from the corners of a 30cm square piece of cardbboard as shown in Figure 1 and bending up the resulting cardboard to form the walls of the box. find the largest possible volume of the box
Let the squares that are cut out each be x by x cm 0<x<15
Now the volume of the prism will be
V=(30−2x)∗(30−2x)∗xV=x(900−120x+4x2)V=4x3−120x2+900xdVdx=12x2−240x+900d2Vdx2=24x−240$Stationarypointswilloccurwhen$dVdx=0 12x2−240x+900=0x2−20x+75=0(x−5)(x−15)=0x=5orx=15
Whenx=5d2Vdx2=24∗5−240<0Maximumatx=5Whenx=15$Volumeis0$$Somaximumvolumeoccurswhenx=5$$Maximumvolume$=V=(30−10)∗(30−10)∗5=2000cm3
Let the squares that are cut out each be x by x cm 0<x<15
Now the volume of the prism will be
V=(30−2x)∗(30−2x)∗xV=x(900−120x+4x2)V=4x3−120x2+900xdVdx=12x2−240x+900d2Vdx2=24x−240$Stationarypointswilloccurwhen$dVdx=0 12x2−240x+900=0x2−20x+75=0(x−5)(x−15)=0x=5orx=15
Whenx=5d2Vdx2=24∗5−240<0Maximumatx=5Whenx=15$Volumeis0$$Somaximumvolumeoccurswhenx=5$$Maximumvolume$=V=(30−10)∗(30−10)∗5=2000cm3