Question

in the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. therefore, animals with a low surface area to volume (sav) ratio tend to retain heat, whereas animals with a high sav ratio (such as children and hummingbirds) lose heat relatively quickly. complete parts (a) through (d). the ellipsoid has an sav ratio of \\(\frac{3}{\sqrt3{24}}\left(\frac{1}{2}+\frac{3}{\sqrt{3}}\sin^{-1}\left(\frac{\sqrt{3}}{2}\
ight)\
ight)\\) (type an equation using \\(r\\) as the variable.) d. graph the sav ratio of the ball of radius \\(r = 1\\) as a function of \\(r\\) (part (b)) and graph the sav ratio of the ellipsoid described in part (c) on the same set of axes. choose the correct graph below. a. graph a description b. graph b description c. graph c description d. graph d description which object has the smaller sav ratio? \\(\circ\\) the ellipsoid \\(\circ\\) the ball

Explanation:

Step1: Recall sphere SAV formula

For a sphere of radius $a$, surface area $S=4\pi a^2$, volume $V=\frac{4}{3}\pi a^3$. SAV ratio: $\frac{S}{V}=\frac{4\pi a^2}{\frac{4}{3}\pi a^3}=\frac{3}{a}$

Step2: Analyze SAV behavior

As $a$ increases, $\frac{3}{a}$ decreases (downward sloping curve). The ellipsoid's SAV ratio is a constant value (given expression has no $a$ variable), so its graph is a horizontal line.

Step3: Match to graph

Only Option A has a downward sloping curve (sphere) and a horizontal line (ellipsoid), with the horizontal line above the curve at large $a$.

Step4: Compare SAV ratios

From the graph, the horizontal line (ellipsoid) lies below the sphere's curve for all $a>1$, meaning the ellipsoid has a smaller SAV ratio.

Answer:

1. Correct graph: Option A
2. Object with smaller SAV ratio: The ellipsoid