Final answer:
An example that shows the surface area to volume relationship is seen in the comparison of different sized cubes and cell biology. As the cell or cube grows, the volume increases exponentially compared to the surface area, decreasing the surface area-to-volume ratio, which affects cellular functions such as nutrient uptake and waste elimination.
Step-by-step explanation:
An example that correctly demonstrates the relationship between surface area and volume can be seen in the context of cell biology. As a cell grows in size, its volume increases faster than its surface area, resulting in a decreased surface area-to-volume ratio. To illustrate, let's consider two cubes, one larger than the other. The small cube with a side length of 1 cm has a surface area of 6 cm² and a volume of 1 cm³, yielding a surface area-to-volume ratio of 6:1. On the other hand, the large cube with a side length of 3 cm has a surface area of 54 cm² and a volume of 27 cm³, leading to a surface area-to-volume ratio of 2:1. This principle is crucial in biology because a high surface area-to-volume ratio is advantageous for cellular processes such as nutrient uptake and waste elimination.
When discussing different shapes, keeping the volume constant, we learn that shape influences surface area. A sphere has the minimum surface area for a given volume compared to other shapes, and elongating the shape in either one or two dimensions increases the surface area. For cylinders, volume is calculated as the cross-sectional area times the height, while surface area includes the areas of the two end-caps plus the side surface area.