Let x------> the length side of the square base of the box y-------> the height of the box
we know that volume of the box=b²*h b=x h=y volume=256 cm³ so 256=x²*y------>y=256/x²--------> equation 1
<span>The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.</span>
surface area of the cardboard=area of the base+perimeter of base*height area of the base=x² perimeter of the base=4*x height=y surface area=x²+4x*y-----> equation 2 substitute equation 1 in equation 2 SA=x²+4x*[256/x²]-----> SA=x²+1024/x
step 1 find the first derivative of SA and equate to zero 2x+1024*(-1)/x²=0------> 2x=1024/x²----> x³=512--------> x=8 cm y=256/x²------> y=256/8²-----> y=4 cm
the answer is the length side of the square base of the box is 8 cm the height of the box is 4 cm
