Largest Area for a Fixed Perimeter

0:00You have 40 metres of fence for a rectangular pen. Which rectangle has the largest area?

0:07The perimeter is the distance around; the area is the space inside. A 5 by 15 pen uses 40 metres and holds 75 square metres.

0:198 by 12 and 10 by 10 also use 40 metres, but give 96 and 100 square metres.

0:27A common mistake is thinking the same perimeter means the same area. It does not.

0:33Call one side x. Length plus width is half of 40, so the other side is 20 minus x.

0:42So A equals x times, in brackets, 20 minus x: 20x minus x squared.

0:50That is a quadratic: x squared is its highest power. Its graph is a curve called a parabola.

0:59Our rectangles are points on it: 5 gives 75, 8 gives 96, and 10 gives 100.

1:07The parabola opens downward, so its vertex, the turning point, is the highest point.

1:14It crosses the x axis at 0 and 20; the vertex is halfway, at 10.

1:20So the largest area comes from a 10 by 10 square: 100 square metres.

1:27Twist: put the pen against a long wall, so the fence covers only three sides.

1:33Two sides are x and the third is 40 minus 2x, so A equals x times, in brackets, 40 minus 2x.

1:43Again it crosses at 0 and 20, so the vertex is at x equals 10.

1:50The third side is 40 minus 2 times 10, which is 20. A 10 by 20 pen holds 200 square metres.

1:59In 3D, among boxes with the same surface area, the cube holds the most volume.

2:06Recap: for a fixed perimeter, the vertex of the area graph gives the largest rectangle, a square.