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The zoo wants to give the elephants as much space as possible in a rectangular enclosure meant for feeding. The zoo has 180 feet of fencing. What should the dimensions of the rectangle be? Be prepared to share your reasoning.
The three prisms shown each have a volume of 216 cubic centimeters. Which prism has the largest surface area? Which has the smallest surface area?
1 cm by 12 cm by 18 cm
9 cm by 8 cm by 3 cm
6 cm by 6 cm by 6 cm
The surface area of the first prism is 492 square centimeters, the surface area of the second is 246 square centimeters, and the surface area of the third is 216 square centimeters. The cube, then, has the smallest surface area. In general, the cube is the rectangular prism with the least amount of surface area for its volume. The 1 by 12 by 18 cm prism has the greatest surface area of the three. If we want a rectangular prism to have more surface area, the best design is to make it wide and long.
Shapes that are more compact, like a cube, have the least surface area for a given volume. But it turns out that spheres do even better than cubes. A sphere with radius 3.72 centimeters, shown in the figure, has an approximate volume of 216 cubic centimeters like the earlier prisms, but its approximate surface area is just 174 square centimeters.
We can see examples of maximizing or minimizing surface area in nature. For instance, a snake is cold-blooded, meaning that it gets its heat from the environment. Its long, narrow shape helps it soak up more heat from the sun. On the other hand, large mammals such as elephants and cows are warm-blooded, which means that they produce their own heat internally. Their shapes are more compact, closer to spheres, and this allows them to lose as little heat as possible through their skin.