Answer:
1.81 inch by 8.38 inch by 6.38 inch
Smallest Value=1.81 inch
Largest Value=8.38 inch
Step-by-step explanation:
The cardboard is 12 in. long and 10 in. wide
Let the length of the square cut off=x
Length of the box=12-2x
Width of the box=10-2x
Height of the box=x
Volume of the box=lwh
V=x(12-2x)(10-2x)
The dimensions of the box that will yield maximum volume occurs at the point where the derivative of V=0.
Thus:
4(30-22x+3x²)=0
Since 4≠0
3x²-22x+30=0
Solving for x using a calculator gives:
x=1.81 or x=5.52
x cannot be 5.52 inch since the width is 10 inch and removing 2(5.52) from the width gives a negative result.
When x=1.81 inch
Length of the box=12-2x=12-2(1.81)=8.38inch
Width of the box=10-2x=10-2(1.81)=6.38 inch
Therefore, the dimensions at which the Volume is maximum are: 1.81 inch by 8.38 inch by 6.38 inch