A box is to have a square base an open top and volume of 2ft Find the dimensions of the box to the nearest inch that uses the least amount of material?
Let the length of the square be x and the height of the box be
h. The volume of the box is the x2h=2ft2 The surface area (remember
the top is open) is x2+4xh Using the first equation, make h the
subject of the formula. h=2/(x2) and substitute this into the
surface area formula. We then have SA=x2+8/x We have a minimum
materials used when the derivative of this equation is equal to 0
SA'(derivative)=2x-8/(x2)=0 2x=8/(x2) 2x3=8 x3=4 x=1.59 ft
h=2/x2=0.79 So the dimensions are 1.59 x 1.59 x 0.79 (2 dec
pl.)