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Ch. 39 - Animal Form and Function
All textbooksFreeman 8th EditionCh. 39 - Animal Form and FunctionProblem 8
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Chapter 38, Problem 8

a. Consider three spheres with radii of 1 cm, 5 cm, and 10 cm. Based on what you read in the chapter, predict which sphere will have the highest surface area to volume ratio, and which sphere will have the lowest.

b. Next, calculate the surface area and the volume of each sphere. (Surface area of a sphere=4πœ‹π‘Ÿ2; volume of a sphere=(4/3)πœ‹π‘Ÿ3.) Plot the results on a graph with radius on the π‘₯-axis and surface area and volume on the 𝑦-axis.

c. Which sphere has the highest surface area to volume ratio? The lowest? Explain how the graph shows the relationship between size and surface area to volume ratio.

d. Now imagine that these spheres represent a small, medium, and large endothermic animal. Which animal would lose heat most rapidly? Explain using the surface area to volume ratio.

Verified step by step guidance
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a. To predict which sphere will have the highest and lowest surface area to volume ratio, recall that as the radius of a sphere increases, the volume increases faster than the surface area. Therefore, the smallest sphere (radius = 1 cm) will likely have the highest surface area to volume ratio, and the largest sphere (radius = 10 cm) will have the lowest.
b. To calculate the surface area and volume for each sphere, use the formulas provided: Surface area = 4Ο€r^2 and Volume = (4/3)Ο€r^3. Calculate these values for each sphere with radii of 1 cm, 5 cm, and 10 cm. Then, plot these values on a graph with the radius on the x-axis and both surface area and volume on the y-axis, using different colors or markers for each to distinguish them.
c. To determine which sphere has the highest and lowest surface area to volume ratio, calculate the ratio for each sphere using the formula: Surface Area to Volume Ratio = Surface Area / Volume. Compare these ratios for all three spheres. The graph will show that as the radius increases, the surface area to volume ratio decreases, illustrating the relationship between size and this ratio.
d. Considering these spheres as models for endothermic animals, the animal represented by the smallest sphere (with the highest surface area to volume ratio) would lose heat most rapidly. This is because a higher surface area to volume ratio means more surface area relative to volume, which facilitates greater heat loss to the environment.
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