Modeling Growth in Whales by Improving the Current Geometric Model

            This study attempts to improve the current geometric model being used to describe growth in whales by quantifying the possible evolutionary forces at work on forming the shape of whales. A lot of information was gathered from the available literature, and many of the calculations and figures were done in Mathematica. Recently, it has been shown that whales fit the model M = k*L^2.8 where M represents mass, L represents length, and k is just some constant coefficient. The two most important forces are believed to be thermoregulation and drag. An increase in surface area for a prolate spheroid leads to a decrease in drag which makes the whale more efficient moving in water, but it also leads to greater heat loss. Looking at the total amount of expended energy as our primary force dictating the shape of a whale, we expect there to be convergence to an optimal value for beta in the model M = k*L^β. This value should be close to 2.8, the value we actually see in whales. The surface area equation for the prolate spheroid can be simplified to 6.28319 x 0.25^β (( 4^{-1 + β} L³)^{1/1 + 2β})^2β + 4.9323 x 0.318471^β ((4^{-1 + 2β} L³)^{1/1 + 2β})^{1 + β}. The current geometric model, M = k*L³, is slightly inaccurate in describing growth in whales. By quantifying and implementing a few evolutionary forces into the model, we can create a more accurate model.