Problem 217 - Calculus Volume 1

Question

The velocity \(V\) (in centimeters per second) of blood in an artery at a distance \(x \mathrm{~cm}\) from the center of the artery can be modeled by the function $$V=f(x)=500\left(0.04-x^{2}\right) \text { for } 0 \leq x \leq 0.2$$ a. Find \(x=f^{-1}(V)\). b. Interpret what the inverse function is used for. c. Find the distance from the center of an artery with a velocity of \(15 \mathrm{~cm} / \mathrm{sec}, 10 \mathrm{~cm} / \mathrm{sec},\) and \(5 \mathrm{~cm} / \mathrm{sec}\)

Step-by-Step Solution

Answer

The inverse function is \( x = \sqrt{0.04 - \frac{V}{500}} \). It finds distance from velocity. Distances: 0.1 cm, 0.141 cm, 0.173 cm.

Step 1: Find the Expression for the Inverse Function

To find the inverse function, start with the given equation for velocity:\[ V = 500(0.04 - x^2) \]Solve for \( x \) in terms of \( V \). First, divide both sides by 500:\[ \frac{V}{500} = 0.04 - x^2 \]Rearrange to solve for \( x^2 \):\[ x^2 = 0.04 - \frac{V}{500} \]Finally, take the square root to solve for \( x \):\[ x = \sqrt{0.04 - \frac{V}{500}} \]This gives us the inverse function \( x = f^{-1}(V) = \sqrt{0.04 - \frac{V}{500}} \).

Step 2: Interpret the Inverse Function

The inverse function \( f^{-1}(V) \) is used to determine the distance \( x \) from the center of the artery given a specific blood velocity \( V \). While the original function \( V=f(x) \) gives the velocity at a given distance, the inverse allows us to find out how far a certain velocity occurs from the center of the artery.

Step 3: Calculate the Distances for Given Velocities

Using the inverse function \( x = \sqrt{0.04 - \frac{V}{500}} \), calculate the distance for each provided velocity:For \( V = 15 \) cm/s:\[ x = \sqrt{0.04 - \frac{15}{500}} = \sqrt{0.04 - 0.03} = \sqrt{0.01} = 0.1 \text{ cm} \]For \( V = 10 \) cm/s:\[ x = \sqrt{0.04 - \frac{10}{500}} = \sqrt{0.04 - 0.02} = \sqrt{0.02} \approx 0.141 \text{ cm} \]For \( V = 5 \) cm/s:\[ x = \sqrt{0.04 - \frac{5}{500}} = \sqrt{0.04 - 0.01} = \sqrt{0.03} \approx 0.173 \text{ cm} \]

Key Concepts

Blood Velocity

Blood velocity refers to the speed at which blood flows through the vessels in our body. It can vary widely depending on where in the circulatory system the measurement takes place. In arteries, blood velocity can be quite high due to the forceful pumping action of the heart.

In the context of this exercise, we are considering a model for blood velocity in an artery based on its distance from the center. The velocity is measured in centimeters per second and changes depending on the distance from the center of the artery.

• Higher at the center and decreases towards the edges.
• This is typical in a circular cross-section of blood vessels where velocity distribution is parabolic.

The given function, and discovering its inverse, allows us to understand the relationship between the speed of blood flow and how far it is from the central point of the artery.

Distance Calculation

The Distance Calculation involves using the inverse function to determine how far away blood traveling at a certain velocity is within the artery. This calculation is vital because knowing both speed and distance is important in medical diagnostics and research.

We use the inverse function \(x = \sqrt{0.04 - \frac{V}{500}}\) to find the distance from the artery's center for a particular velocity. Let's break it down:

• For blood moving at 15 cm/s, it's 0.1 cm away from the center.
• For 10 cm/s, it's approximately 0.141 cm away.
• At 5 cm/s, the distance is about 0.173 cm from the center.

These calculations are necessary for visualizing blood flow, helping medical professionals identify any irregularities in circulation patterns that might require attention.

Function Modeling

Function modeling is about using mathematical equations to represent real-world scenarios, in this case, blood flow. The function provided in the exercise allows us to model the velocity of blood as it travels through an artery based on the distance from its center.

• The original function, \(V=f(x) = 500(0.04-x^2)\), models how velocity decreases as we move further from the center.
• In this model, the coefficient 500 represents how fast blood flows when unimpeded, while the term \(0.04-x^2\) represents diminishing velocity as we move along the artery.

Understanding this model helps predict how variations in artery diameter or blockages could affect blood velocity. By analyzing these changes, one can get insights into health issues such as artery blockages or the overall efficiency of the circulatory system. The concept of inverse functions is critical for retracing such changes back to their causes, offering a comprehensive insight into dynamic systems such as blood circulation.