Problem 80 - Brief Applied Calculus

Question

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 mos 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-to-shoulder length can be measured from their skeletons: Triceratops, whose hip-to-shoulder length was 14 feet.

Step-by-Step Solution

Answer

The average body thickness of the Triceratops is approximately 20.98 feet.

Step 1: Understand the given formula

The formula provided for average body thickness is \( T = 0.4 imes (L)^{3/2} \), where \( T \) is the average body thickness, and \( L \) is the hip-to-shoulder length in feet. In this problem, we need to calculate \( T \) for the dinosaur with a given \( L \).

Step 2: Substitute the hip-to-shoulder length into the formula

We know that the hip-to-shoulder length \( (L) \) of the Triceratops is 14 feet. Substitute \( L = 14 \) into the formula: \[ T = 0.4 imes (14)^{3/2} \]

Step 3: Calculate the expression \( (14)^{3/2} \)

First, find the square root of 14 and then cube it: \[ (14)^{3/2} = (\sqrt{14})^3 \]Calculate \( \sqrt{14} \approx 3.74 \). Now cube this result:\[ (3.74)^3 \approx 52.44 \]

Step 4: Multiply by 0.4

Now multiply the result from Step 3 by 0.4 as per the formula:\[ T = 0.4 imes 52.44 \approx 20.976 \]

Step 5: Round to a sensible precision

Typically, one would round to two decimal places for such measurements. Hence, round 20.976 to 20.98 feet for the average body thickness.

Key Concepts

Power Law

In mathematics and physics, a power law is a functional relationship between two quantities, where one quantity varies as a power of another. This type of relationship is often expressed in the form of a mathematical equation: \( y = ax^b \). Here, \( y \) and \( x \) are the variables, \( a \) is a constant, and \( b \) is the exponent.

Power laws are useful for modeling how different characteristics scale with each other, such as size various animal body parts or environmental phenomena.

• Example: In biology, they can model how metabolic rates change with body size across species.
• They help identify proportional relationships where doubling one variable might result in a less or more than doubling effect on another.

This kind of mathematical relationship is simple yet powerful, enabling deep insights across various fields from ecology to market analysis.

Exponents

Exponents are a mathematical notation that indicates the number of times a number or expression is multiplied by itself. For example, in the expression \( x^3 \), \( 3 \) is the exponent, meaning \( x \) is multiplied by itself three times: \( x \times x \times x \).

Exponents can be whole numbers, fractions, or even negative numbers, and they make it easier to represent large and small numbers concisely.

• Fractional Exponents: They denote both root and power simultaneously. For instance, \( x^{3/2} \) means the square root of \( x \) raised to the third power.
• Importance: They simplify computations and formula representations, which is crucial in scientific calculations and model building.

Understanding how to work with exponents, especially fractional ones, is key in fields like physics, chemistry, and biology.

Body Measurements

Body measurements refer to the dimensions that are used to describe the size and shape of living organisms. In allometry, which studies the relationship of body parts, specific measurements are crucial. For instance, scientists often use measurements like:

• Hip-to-shoulder length: This measures the distance between an animal's hip and shoulder joints.
• Body thickness: Typically measured vertically, indicating how thick or robust an animal's body is at a given point.

For accurate allometric studies, measurements need to be precise, as they're used to model relationships like those expressed in power laws. These measurements aid in understanding the physical capabilities and adaptations of organisms in their natural environments.

Dinosaurs

Dinosaurs are an extinct group of reptiles that roamed the Earth during the Mesozoic Era. They vary vastly in size, shape, and lifestyle.

Much of our knowledge about dinosaurs comes from fossil records, which provide clues about their biology, evolution, and the ecosystems they inhabited. Notable points include:

• Diversity: Dinosaurs were incredibly diverse, with species ranging from tiny bird-like creatures to enormous long-necked giants.
• Fossil Evidence: These include bones, footprints, and even soft tissues, which help scientists reconstruct their appearance and behaviors.

This information helps paleontologists (scientists who study fossils) piece together Earth's history, understanding how these ancient creatures lived and ultimately met their demise.

Triceratops

The Triceratops is one of the most iconic dinosaurs, known for its three distinctive facial horns and large frill. Native to what is now North America, it lived approximately 68 to 66 million years ago during the late Cretaceous Period.

Often depicted as a herbivorous dinosaur, the Triceratops had:

• Robust Build: Measuring up to 30 feet in length, with a hip-to-shoulder length providing insights into its body proportions.
• Defensive Features: It is believed these horns and frill were used to fend off predators, possibly in combination with social behaviors.

Triceratops remains a key subject of study in paleontology, with its fossilized remains offering a wealth of information about dinosaur biology and the Mesozoic ecosystems.