Final answer:
The assertion that surface area and volume increase at the same rate is false. As a cell grows, the surface area increases by the square of the radius, while the volume increases by the cube, decreasing the surface area-to-volume ratio.
Step-by-step explanation:
The statement that as the surface area of a body increases, its volume increases at the same rate is false. An important concept in biology is the surface area-to-volume ratio, which has significant implications for cells and their functions. As a cell grows, its volume increases faster than its surface area. Considering a sphere, the formula for the surface area (SA) is 4πr², where r is the radius, and the formula for volume (V) is (4/3)πr³. Therefore, as the cell's radius increases, SA increases by the square of the radius, while V increases by the cube of the radius, leading to a decrease in the SA:V ratio. This can result in insufficient surface area to support the cell's needs if it grows too large. As depicted in Figure 4.7, the smaller cell has a volume of 1 mm³ and a surface area of 6 mm² (SA:V ratio of 6:1) while the larger cell has a volume of 8 mm³ and a surface area of 24 mm², resulting in a reduced SA:V ratio of 3:1.