Problem 54 - Calculus for Biology and Medicine

Question

In each of the questions 54-56, use a measured power law to make predictions. Estimating the mass of extinct animals is very difficult, since estimates have to be made based on fossil skeletons and without knowledge of the muscles and other tissues that make up much of an animal's mass. van Valkenburgh (1990) measured body mass and skull length in living carnivores and found that, if \(y\) is body mass measured in \(\mathrm{kg}\) and \(x\) is skull length measured in \(\mathrm{mm}\), $$ y=k x^{3.13} \text { . } $$ (a) A gray wolf ( Canis lupus) weighs \(45 \mathrm{~kg}\), and has a skull length of \(275 \mathrm{~mm} .\) Calculate the coefficient of proportionality \(k\) in the power law. (b) Estimate the weight of the sabretooth tiger (Smilodon populator \() .\) S. populator fossils have skull lengths of \(350 \mathrm{~mm}\). (c) How strong was Smilodon's bite? Hartstone-Rose, Perry, \mathrm{\\{} a n d ~ M o r r o w ~ ( 2 0 1 2 ) ~ s t u d i e d ~ t h e ~ c o r r e l a t i o n ~ b e t w e e n ~ b i t e ~ f o r c e ~ and body mass. They found that, if bite force (measured in newtons, or \(\mathrm{N}\) ) is \(f\) and body mass (measured in \(\mathrm{kg}\) ) is \(y\), then $$ f=c y^{0.96} $$ for some constant \(c\). (i) The tiger has a bite force of \(7980 \mathrm{~N}\), and a mass of \(200 \mathrm{~kg}\). Calculate the constant \(c\). (ii) Estimate the bite force of Smilodon populator using your body mass estimate from part (b).

Step-by-Step Solution

Answer

(a) \(k \approx 1.99 \times 10^{-6}\); (b) weight: \(110\) kg; (c) bite force: \(4402\) N.

Step 1: Derive the value of k

To find the coefficient of proportionality \(k\), we use the wolf's measurements: body mass \(y = 45\) kg and skull length \(x = 275\) mm. Substitute these values into the power law equation: \(45 = k (275)^{3.13}\). Solving for \(k\) gives us \(k = \frac{45}{(275)^{3.13}}\).

Step 2: Calculate k

Using a calculator, compute \((275)^{3.13} \approx 22599798.69\). Hence, \(k = \frac{45}{22599798.69} \approx 1.99 \times 10^{-6}\).

Step 3: Estimate the mass of the sabretooth tiger

Substitute the skull length of Smilodon populator (\(x = 350\) mm) into the equation \(y = k x^{3.13}\) with the previously calculated \(k\): \(y = 1.99 \times 10^{-6} (350)^{3.13}\).

Step 4: Compute the mass of Smilodon populator

Compute \((350)^{3.13} \approx 55096347.67\). Thus, \(y = 1.99 \times 10^{-6} \times 55096347.67 \approx 110\) kg.

Step 5: Derive the value of c

Using the tiger's bite force: \(f = 7980\) N and body mass \(y = 200\) kg, substitute in the equation \(f = c y^{0.96}\): \(7980 = c (200)^{0.96}\). Solve for \(c\) to get \(c = \frac{7980}{(200)^{0.96}}\).

Step 6: Calculate c

Using a calculator, \((200)^{0.96} \approx 192.42\). Thus, \(c = \frac{7980}{192.42} \approx 41.48\).

Step 7: Estimate Smilodon's bite force

Use the estimated mass of Smilodon \(y = 110\) kg in \(f = c y^{0.96}\) with the previously calculated \(c = 41.48\): \(f = 41.48 \times (110)^{0.96}\).

Step 8: Compute the bite force of Smilodon

Compute \((110)^{0.96} \approx 106.11\). Therefore, \(f = 41.48 \times 106.11 \approx 4401.64\) N.

Key Concepts

Body Mass Estimation

Estimating the body mass of extinct animals like the Smilodon populator can be quite challenging. Scientists often rely on fossil evidence combined with mathematical models to make educated guesses. One such method involves using a power law, which relates different physical characteristics through an equation. In this case, a measured power law relates the skull length of an animal to its body mass. The power law formula used is \( y = kx^{3.13} \), where \( y \) stands for the body mass in kilograms, \( x \) is the skull length in millimeters, and \( k \) is the coefficient of proportionality.
This formula implies that as the skull length increases, the body mass does not simply increase linearly, but rather scales with the power of 3.13. This exponent indicates how much influence the skull size has on estimating the body mass. By plugging in the skull length of Smilodon populator into this equation, we can estimate its weight as approximately 110 kg. This approach gives paleontologists a valuable tool in paleobiology, allowing them to make informed estimates on the features of animals that lived millions of years ago.

Coefficient of Proportionality

The coefficient of proportionality, represented by \( k \), plays a crucial role in power law predictions like our body mass estimation. Think of it as a scaling factor that adjusts the basic relationship between skull length and body mass. It accounts for the unit differences and ensures that our equation relates the two quantities accurately.
In the case of the gray wolf, with a known weight and skull length, we can determine \( k \) by substituting these values into the power law formula. By doing so, we can mathematically derive the equation:

• For a gray wolf with a body mass \( y = 45 \) kg and skull length \( x = 275 \) mm, the process is as follows: \( 45 = k \times (275)^{3.13} \).
• This gives us \( k = \frac{45}{(275)^{3.13}} \approx 1.99 \times 10^{-6} \).

You can see how \( k \) effectively acts as a bridge, making sure that our formula accurately scales to the measurements we have. Without the coefficient of proportionality, our model wouldn't accurately reflect the real-world data.

Bite Force Calculation

When attempting to determine the bite force of an extinct animal like Smilodon populator, we lean again on the concept of power laws. This time, we relate the bite force (\( f \)) to the body mass (\( y \)) through the formula: \( f = c y^{0.96} \). Here, \( c \) is a constant that captures specific characteristics, like jaw mechanics and muscle efficiency.
To calculate \( c \), scientists use known values from similar living species. For example, in the exercise, they use a tiger's known bite force and body mass. The example given shows a tiger with a bite force of \( 7980 \) N and a mass of \( 200 \) kg, leading to:

• \( 7980 = c \times (200)^{0.96} \)
• This gives us \( c = \frac{7980}{(200)^{0.96}} \approx 41.48 \)

Once calculated, \( c \) can be used to estimate the bite force of Smilodon populator by substituting its estimated body mass. Here, utilizing its estimated mass of 110 kg, we find:

• \( f = 41.48 \times (110)^{0.96} \approx 4401.64 \) N

These calculations give us insights into the power dynamics of extinct predators and help us understand their hunting and feeding behaviors.