Problem 51 - Calculus for Biology and Medicine

Question

Panting in Animals Animals use different strategies to control their internal temperature depending on how hot they are. When the core temperature of a dog, duck, or cat exceeds a critical value, it will start to pant (make quick, gasping breaths that increase evaporation of water from the tongue and mouth). Vieth (1989) studied heat loss as a function of the ducks' core temperature, \(T\). She found that different functions described heat loss below the temperature at which the ducks started to pant and above this temperature. If \(H(T)\) is the rate of heat loss: $$ H(T)=\left\\{\begin{array}{ll} 0.6 & \text { if } T \leq T_{c} \\ 4.3 T-183 & \text { if } T>T_{c} \end{array}\right. $$ (here \(T\) is measured in \({ }^{\circ} \mathrm{C}\) and \(H(T)\) in watts per \(\mathrm{kg}\) of body mass \()\) (a) Calculate the value of \(T_{c}\) that makes \(H(T)\) continuous for all \(T\). (b) Draw the graph of the function \(H(T)\) over the normal body temperature range for ducks: \(41^{\circ} \mathrm{C} \leq T \leq 44^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) \(T_c \approx 42.7^{\circ} C\). (b) Graph is a horizontal line at 0.6 until \(42.7^{\circ} C\), then a line following \(4.3T - 183\).

Step 1: Understanding Continuity

To ensure that the function \(H(T)\) is continuous, \(H(T)\) should not have any jumps or breaks at \(T = T_c\). This means the value of \(H(T)\) from the first piece must equal the value from the second piece at \(T = T_c\). Thus, we set \(H(T)\) from the first part equal to \(H(T)\) from the second part.

Step 2: Setting Up the Equation for Continuity

We set the two parts of the piecewise function equal to each other at \(T = T_c\) to ensure continuity:\[0.6 = 4.3T_c - 183\]

Step 3: Solving for \(T_c\)

Solve the equation from Step 2. Add 183 to both sides to isolate the term with \(T_c\):\[0.6 + 183 = 4.3T_c\]Calculate the left side:\[183.6 = 4.3T_c\]Divide both sides by 4.3 to solve for \(T_c\):\[T_c = \frac{183.6}{4.3}\]\(T_c \approx 42.7^{\circ} \mathrm{C}\).

Step 4: Drawing the Graph

For \(41^{\circ} C \leq T \leq 44^{\circ} C\):1. For \(T \leq T_c \), \(H(T) = 0.6\).2. For \(T > T_c \approx 42.7\), use \(H(T) = 4.3T - 183\).Draw a horizontal line at \(H(T) = 0.6\) from \(41^{\circ} C\) to \(42.7^{\circ} C\).After \(42.7^{\circ} C\), draw the line of \(H(T) = 4.3T - 183\). Calculate a couple of points to sketch the line accurately if needed.

Key Concepts

Temperature Regulation in Animals

Animals have developed various strategies to maintain their internal temperature, a process crucial for survival. This process is termed thermoregulation.
Many animals rely on physiological and behavioral adaptations to manage body temperature effectively. For instance, when a dog, duck, or cat experiences temperatures beyond their normal range, they might start panting.
Panting is a common mechanism to facilitate heat loss. Rapid, shallow breathing increases the evaporation of water from the surfaces of the tongue and mouth, cooling the body.

• This form of evaporative cooling is especially effective under low humidity conditions.
• It's a quick response to prevent overheating and maintain a stable internal environment.

Understanding how animals regulate their temperature helps in studying ecological adaptations and also informs conservation strategies.

Piecewise Functions

Piecewise functions allow us to model situations where a rule or relationship changes based on input values. For example, in analyzing the heat loss in ducks, the function governing this loss alters depending on whether the duck is panting or not.
The piecewise function used in the exercise is defined in two parts:

• Below the critical temperature (T_c), the heat loss is constant (H(T) = 0.6).
• Above T_c, heat loss increases linearly (H(T) = 4.3T - 183).

Piecewise functions are valuable in biological studies where physiological responses can vary dramatically under different conditions. They help describe systems that are too complex for a single formula.

Heat Transfer Analysis

In biology, understanding heat transfer is key to interpreting how animals interact with their environment. Heat transfer involves the mechanisms by which heat flows from or into an animal's body.
There are three main forms of heat transfer:

• Conduction - Transfer of heat through direct contact.
• Convection - Heat transfer through fluid movement around the body, like air or water currents.
• Radiation - Loss of heat in the form of infrared rays.

The rate of heat loss in ducks, described by the piecewise function, is part of a larger analysis of how efficient different heat transfer methods help maintain body temperature. Recognizing these connections aids in constructing comprehensive models of animal physiology.

Continuity in Calculus

Continuity in calculus refers to how smooth a function's graph is. For a function to be continuous at a point, it must not have any breaks or jumps.
In the given problem, ensuring continuity of the function representing heat loss at the critical temperature T_c is vital:

• The two equations describing heat loss must intersect smoothly at T_c (no sudden jumps).
• This means both parts of the piecewise function need to give the same output when T = T_c.

Establishing continuity is essential in creating realistic models by ensuring there's a seamless transition from one physiological response to another. This approach not only aids in accurate predictions but also in troubleshooting potential anomalies in datasets.