ALLOMETRY: Dinosaurs The study of size and shape is called...

ALLOMETRY: Dinosaurs The study of size and shape is called "allometry," and many allometric relationships involve exponents that are fractions or decimals. For example, the body measurements of most four-legged animals, from mice to elephants, obey (approximately) the following power law: $$\left(\begin{array}{c} \text { Average body } \\ \text { thickness } \end{array}\right)=0.4 \text { (hip-to-shoulder length) }^{3 / 2}$$ where body thickness is measured vertically and all measurements are in feet. Assuming that this same relationship held for dinosaurs, find the average body thickness of the following dinosaurs, whose hip-toshoulder length can be measured from their skeletons: Triceratops, whose hip-to-shoulder length was 14 feet.

ALLOMETRY: Dinosaurs The study of size and shape is called "allometry," and many allometric relationships involve exponents that are fractions or decimals. For example, the body measurements of most four-legged animals, from mice to elephants, obey (approximately) the following power law:
$$\left(\begin{array}{c}
\text { Average body } \\
\text { thickness }
\end{array}\right)=0.4 \text { (hip-to-shoulder length) }^{3 / 2}$$
where body thickness is measured vertically and all measurements are in feet. Assuming that this same relationship held for dinosaurs, find the average body thickness of the following dinosaurs, whose hip-toshoulder length can be measured from their skeletons:
Triceratops, whose hip-to-shoulder length was 14 feet.

Key Concepts

Allometry

Allometry is the study of the relationship between the size of an organism and the shape, anatomy, physiology, and behavior of its parts. In biological studies, this concept helps explain how different traits scale nonlinearly with body size and why these relationships are observed across a wide range of species. It is fundamental to understanding the variations in body proportions as animals grow or as sizes vary among species.

Power Law Relationships

Power law relationships are mathematical expressions where one quantity varies as a power of another. In biological allometry, these relationships express how traits such as body dimensions scale with each other, typically in the form of Y = aX^b. This formula captures non-linear scaling, meaning that a doubling of one measurement does not necessarily result in a doubling of another, but rather an increase determined by the exponent.

Scaling Exponents

Scaling exponents are the powers in power law relationships that determine how one variable changes in relation to another. They often appear as fractions or decimals, indicating non-proportional scaling between measurements. These exponents are crucial in allometry as they quantify the rate at which one physical trait grows compared to another, providing insight into biological growth patterns and structural constraints.

Dimensional Consistency

Dimensional consistency involves ensuring that the units of measurement used in allometric equations are coherent and correctly applied. In the context of power law relationships, maintaining consistent units (such as feet for length measurements) is essential for the validity of the model and the comparisons made between different organisms or across different scales.

Transcript

00:01 Okay, number 42.

00:04 The problem is dinosaur running.

00:07 Oh, interesting.

00:09 An article titled how dinosaurs ram explains that the local motion of different -sized animals can be compared when they have the same front number defined as f equals v squared divided by g multiplied l, where v is the velocity, g is the acceleration of gravity.

00:35 Now it's a constant here, which is 9 .81 m per second square, and l is the length in meters.

00:47 Okay, a1 resort described in the article is that different animals change from a crote to gallop, the same fraud lumber roughly 2 .56.

01:06 Okay, so we have the same fraud lumber, 2 .56.

01:14 Now find the velocity at which this change occurs from a ferret with a lagnance of 0 .09 and a rhinoceros, okay, with a lagoonine, of 1 .2m.

01:35 So we got f equals 2 .56 which is a constant here in this situation f is 2 .56.

01:47 We want to know the v where we have the g is 9 .81 and we have the likeness for those two different animals so we just need plugging for each one.

01:59 Okay, so a, we have f equals v squared divided by g l.

02:08 Now for ferret f is 2 .56, g is 9 .81.

02:23 Ferret.

02:26 His leg means is short, 0 .0 line.

02:32 We plug in, we got 2 .56 equals.

02:37 9 .81, multiply 0 .09, we square.

02:44 Now we in the speed we want to know.

02:51 Now we square equals, okay, let's pull out our calculator.

02:57 Just 2 .56, multiply 9 .81, multiply 0 .09 equals.

03:09 Now we take a square root.

03:11 So we, okay, we square is 2 .26.

03:21 We eat roughly 1 .503 meters per second.