Problem 52
Question
How we cough a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity \(v\) can be modeled by the equation $$ v=c\left(r_{0}-r\right) r^{2} \mathrm{cm} / \mathrm{sec}, \quad \frac{r_{0}}{2} \leq r \leq r_{0} $$ where \(r_{0}\) is the rest radius of the trachea in centimeters and \(c\) is a positive constant whose value depends in part on the length of the trachea. Show that \(v\) is greatest when \(r=(2 / 3) r_{0},\) that is, when the trachea is about 33\(\%\) contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. b. Take \(r_{0}\) to be 0.5 and \(c\) to be 1 and graph \(v\) over the interval \(0 \leq r \leq 0.5\) . Compare what you see with the claim that \(v\) isat a maximum when \(r=(2 / 3) r_{0} .\)
Step-by-Step Solution
Verified Answer
The velocity \( v \) is maximized when \( r = \frac{2}{3} r_0 \), i.e., when the trachea contracts about 33%.
1Step 1: Identify the Function
We are given the function for the average flow velocity \( v \):\[ v = c (r_0 - r) r^2 \] where \( r_0 \) is a constant and \( r \) is the variable representing the radius of the trachea.
2Step 2: Determine the Range
The function is defined in the range \( \frac{r_0}{2} \leq r \leq r_0 \). We need to analyze the function within this interval, where the rest radius \( r_0 \) is the maximum radius when the trachea is not contracted.
3Step 3: Find the Critical Points
To find where the velocity \( v \) is maximized, we first find the derivative \( v'(r) \) and set it to zero:\[ v'(r) = \frac{d}{dr}[c(r_0 - r)r^2] = c\left[(r_0 - r)(2r) + (-1)r^2 \right] = c(2r_0r - 3r^2) \]Set \( v'(r) = 0 \):\[ 2r_0r - 3r^2 = 0 \]Factor the equation:\[ r(2r_0 - 3r) = 0 \] This gives \( r = 0 \) or \( r = \frac{2}{3}r_0 \).
4Step 4: Verify Maximum
Since \( r = \frac{2}{3}r_0 \) is within the given interval \( \left[ \frac{r_0}{2}, r_0 \right] \), check if it is a maximum using the second derivative test:\[ v''(r) = \frac{d}{dr}[c(2r_0r - 3r^2)] = c(2r_0 - 6r) \]Evaluate \( v''\left(\frac{2}{3}r_0\right) \):\[ v''\left(\frac{2}{3}r_0\right) = c(2r_0 - 6 \cdot \frac{2}{3}r_0) = -2cr_0 \]Since \( v''(r) < 0 \), the function reaches a maximum at \( r = \frac{2}{3}r_0 \).
5Step 5: Graph the Function
We take \( r_0 = 0.5 \) and \( c = 1 \), and then analyze the function \( v = (0.5 - r)r^2 \) in the interval \( 0 \leq r \leq 0.5 \).Graphing \( v \) demonstrates that it reaches a peak when \( r = \frac{2}{3} \times 0.5 \approx 0.333 \), confirming the maximum where the trachea is about 33% contracted.
6Step 6: Conclusion
The critical point \( r = \frac{2}{3}r_0 \) maximizes the flow velocity, aligning precisely with the maximum velocity claim at about 33% contraction of the trachea, as can be visually confirmed from the graph.
Key Concepts
Trachea ContractionFlow Velocity OptimizationSecond Derivative TestGraphing Functions
Trachea Contraction
When we cough, our trachea, or windpipe, contracts. This is an involuntary reflex aiming to maximize the speed of the expelled air, facilitating the clearance of irritants from our respiratory system. What's fascinating is that this contraction process is highly optimized and indicative of a delicate balance struck by our body.
When the trachea contracts, the radius of its inner tube reduces. In the context of the given exercise, we deal with a model that measures average flow velocity, represented as a function of how contracted the trachea is. The trachea doesn't need to close too much—about 33% contraction is optimal for achieving maximum velocity.
This contraction is measured in terms of a reduced radius, denoted by \( r \), as a proportion of its original resting radius, \( r_0 \). Contraction to \( \frac{2}{3} r_0 \) indicates that it might appear only modestly closed, but this degree is crucial for efficiently blowing air out during a cough.
When the trachea contracts, the radius of its inner tube reduces. In the context of the given exercise, we deal with a model that measures average flow velocity, represented as a function of how contracted the trachea is. The trachea doesn't need to close too much—about 33% contraction is optimal for achieving maximum velocity.
This contraction is measured in terms of a reduced radius, denoted by \( r \), as a proportion of its original resting radius, \( r_0 \). Contraction to \( \frac{2}{3} r_0 \) indicates that it might appear only modestly closed, but this degree is crucial for efficiently blowing air out during a cough.
Flow Velocity Optimization
In the context of coughing, maximizing the flow velocity is vital. This optimization is crucial for expelling air at a speed that effectively clears out obstructions or irritants within the trachea.
To achieve this maximum velocity, one needs to carefully adjust the contraction of the trachea or the decrease in radius \( r \). The equation for the average flow velocity is given by \( v = c(r_0 - r) r^2 \), where \( c \) is a positive constant.
By setting the first derivative of this function with respect to \( r \) to zero, we find critical points that indicate possible maxima or minima. Among these, the point \( r = \frac{2}{3} r_0 \) stands out as it provides the maximum velocity. This tells us that the optimal contraction is about 33%, ensuring efficient air propulsion.
To achieve this maximum velocity, one needs to carefully adjust the contraction of the trachea or the decrease in radius \( r \). The equation for the average flow velocity is given by \( v = c(r_0 - r) r^2 \), where \( c \) is a positive constant.
By setting the first derivative of this function with respect to \( r \) to zero, we find critical points that indicate possible maxima or minima. Among these, the point \( r = \frac{2}{3} r_0 \) stands out as it provides the maximum velocity. This tells us that the optimal contraction is about 33%, ensuring efficient air propulsion.
Second Derivative Test
The Second Derivative Test is an effective method used to determine whether a critical point found in a function corresponds to a maximum, a minimum, or a point of inflection.
After identifying the critical points via the first derivative, calculating the second derivative \( v''(r) \) of the velocity function provides further insights. If \( v''(r) < 0 \) at a critical point, the function is concave down at that point, indicating a local maximum.
In this exercise, the second derivative is \( v''(r) = c(2r_0 - 6r) \). Evaluating it at the critical point \( r = \frac{2}{3}r_0 \) shows a negative value, confirming that the velocity function achieves a maximum here. This fundamental calculus concept supports our understanding of how optimal contraction leads to maximum cough velocity.
After identifying the critical points via the first derivative, calculating the second derivative \( v''(r) \) of the velocity function provides further insights. If \( v''(r) < 0 \) at a critical point, the function is concave down at that point, indicating a local maximum.
In this exercise, the second derivative is \( v''(r) = c(2r_0 - 6r) \). Evaluating it at the critical point \( r = \frac{2}{3}r_0 \) shows a negative value, confirming that the velocity function achieves a maximum here. This fundamental calculus concept supports our understanding of how optimal contraction leads to maximum cough velocity.
Graphing Functions
Graphing functions is a visual way to understand mathematical concepts and verify analytical solutions.
For this exercise, plotting the velocity function helps showcase the behavior of the tracheal air velocity across the permissible radius range \( 0 \leq r \leq 0.5 \).
By setting \( r_0 = 0.5 \) and \( c = 1 \), and graphing \( v(r) = (0.5 - r)r^2 \), the curve's peak at around \( r = 0.333 \) demonstrates where maximum velocity occurs, reinforcing that the trachea contracts to about 33% for optimal airflow during coughing.
For this exercise, plotting the velocity function helps showcase the behavior of the tracheal air velocity across the permissible radius range \( 0 \leq r \leq 0.5 \).
By setting \( r_0 = 0.5 \) and \( c = 1 \), and graphing \( v(r) = (0.5 - r)r^2 \), the curve's peak at around \( r = 0.333 \) demonstrates where maximum velocity occurs, reinforcing that the trachea contracts to about 33% for optimal airflow during coughing.
- The graph helps visualize the otherwise abstract mathematical concepts.
- It confirms analytic calculations and helps identify behavior over the defined interval.
- Graphing serves as double-proof, aligning theoretical findings with practical representation.
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