Problem 79
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: Diplodocus, whose hip-to-shoulder length was 16 feet.
Step-by-Step Solution
Verified Answer
The average body thickness of a Diplodocus is 25.6 feet.
1Step 1: Understand the Power Law Formula
The given power law formula is \( \text{Average body thickness} = 0.4 \times (\text{hip-to-shoulder length})^{3/2} \). This formula will be used to find the average body thickness by substituting the dinosaur's hip-to-shoulder length.
2Step 2: Substitute the Hip-to-Shoulder Length
For a Diplodocus, the hip-to-shoulder length provided is 16 feet. Substitute 16 for the hip-to-shoulder length in the formula: \( \text{Average body thickness} = 0.4 \times (16)^{3/2} \).
3Step 3: Calculate the Exponent
Calculate \((16)^{3/2}\). First, find the square root of 16, which is 4. Then raise 4 to the power of 3: \((4)^3 = 64\).
4Step 4: Calculate the Average Body Thickness
Use the result from Step 3 in the power law formula to find the average body thickness: \( \text{Average body thickness} = 0.4 \times 64 \). Multiply these values to get 25.6.
Key Concepts
Power LawExponentsBody MeasurementsHip-to-Shoulder Length
Power Law
Power laws are mathematical relationships where one quantity varies as a power of another. These laws are characterized by a formula in which two quantities are expressed by a constant multiplier and an exponent.
In this exercise, the formula given is an example of a power law, describing how the average body thickness of animals can be estimated from their hip-to-shoulder length.
The relationship is expressed as follows: \[ \text{Average body thickness} = 0.4 \times (\text{hip-to-shoulder length})^{3/2} \] This formula shows that the body thickness doesn’t increase linearly with hip-to-shoulder length but follows a specific power relation.
Power laws like these are common in allometry, highlighting how biological growth often involves non-linear scaling.
In this exercise, the formula given is an example of a power law, describing how the average body thickness of animals can be estimated from their hip-to-shoulder length.
The relationship is expressed as follows: \[ \text{Average body thickness} = 0.4 \times (\text{hip-to-shoulder length})^{3/2} \] This formula shows that the body thickness doesn’t increase linearly with hip-to-shoulder length but follows a specific power relation.
Power laws like these are common in allometry, highlighting how biological growth often involves non-linear scaling.
Exponents
Exponents are used to indicate how many times a number, known as the base, is multiplied by itself. In our context, the base is the measurement of hip-to-shoulder length, and the exponent is \(\frac{3}{2}\).
Understanding how to work with exponents is crucial for applying the power law formula. For the Diplodocus, we calculate the exponent as follows:
Exponents simplify representing repeated multiplication, especially with fractional or decimal powers, making them highly useful in allometric analyses.
Understanding how to work with exponents is crucial for applying the power law formula. For the Diplodocus, we calculate the exponent as follows:
- Find the square root: \(\sqrt{16} = 4\).
- Raise the result to the power of 3: \(4^3 = 64\).
Exponents simplify representing repeated multiplication, especially with fractional or decimal powers, making them highly useful in allometric analyses.
Body Measurements
Body measurements such as body thickness and hip-to-shoulder length are vital in understanding the physical characteristics of animals. These measurements help compare individuals or species and to interpret growth patterns or adaptations.
In the given exercise, body thickness is derived from hip-to-shoulder length using a power law. This approach allows researchers to estimate characteristics that might be difficult to measure directly, particularly in extinct animals such as dinosaurs.
In the given exercise, body thickness is derived from hip-to-shoulder length using a power law. This approach allows researchers to estimate characteristics that might be difficult to measure directly, particularly in extinct animals such as dinosaurs.
Hip-to-Shoulder Length
The hip-to-shoulder length is a key measurement in allometry and serves as the base in the power law formula for estimating body thickness.
This measurement is often used because it’s a straightforward metric that strongly correlates with other body sizes. For a Diplodocus, with a hip-to-shoulder length of 16 feet, it illustrates how different prehistoric creatures could be analyzed.
By substituting this length in our formula, we computed the average body thickness, providing insights into the physical stature of such dinosaurs.
Thus, the hip-to-shoulder length acts as a practical indicator to decode and understand the size and scale relative to other body measurements.
This measurement is often used because it’s a straightforward metric that strongly correlates with other body sizes. For a Diplodocus, with a hip-to-shoulder length of 16 feet, it illustrates how different prehistoric creatures could be analyzed.
By substituting this length in our formula, we computed the average body thickness, providing insights into the physical stature of such dinosaurs.
Thus, the hip-to-shoulder length acts as a practical indicator to decode and understand the size and scale relative to other body measurements.
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