Maximizing the volume of an open box

0:00A square sheet of cardboard is 20 centimetres on each side.

0:05Cut a square of side x from each corner.

0:08Fold up the sides to make an open box, a box with no lid.

0:13Each base edge loses x at both ends, so it is 20 minus 2x. The height is x.

0:22Volume is length times width times height, so V equals x times 20 minus 2x, squared.

0:31Optimization means finding the best value. Which x gives the largest volume?

0:37The domain is the values of x that make sense: x and 20 minus 2x must be more than 0, so x is between 0 and 10.

0:48If x is 2, the base is 16, so V is 2 times 16 squared, which is 512.

0:57Then 3 gives 588, 4 gives 576, and 5 gives 500.

1:05The volume goes up, then down, so the largest is between 2 and 4.

1:11Now graph V against x with technology, such as a graphing calculator.

1:17The highest point is the maximum: x is about 3.33, and V is about 592.6.

1:26A common mistake is to pick a tiny or a huge cut, but x equals 1 gives only 324, and 9 gives only 36.

1:37Check: the base is 13.34, and 3.33 times 13.34 squared is about 592.6. So the best cut is about 3.33 centimetres.

1:52To optimize, write a formula, find the domain, graph it, and read off the highest point.