Problem 90 - Calculus and its applications

Question

The equation $$ S(r)=\frac{1}{r^{4}} $$ can be used to determine the resistance to blood flow, \(S\). of a blood vessel that has radius \(r\), in millimeters (mm). a) Find the rate of change of resistance with respect to \(r\), the radius of the blood vessel. b) Find the resistance at \(r=1.2 \mathrm{~mm}\). c) Find the rate of change of \(S\) with respect to \(r\) when \(r=0.8 \mathrm{~mm}\)

Step-by-Step Solution

Answer

a) Rate of change: \( S'(r) = -\frac{4}{r^5} \). b) Resistance at \( r = 1.2 \) mm: \( \approx 0.482 \). c) Rate of change at \( r = 0.8 \) mm: \( \approx -9.77 \).

Step 1: Understanding the Function

The function given is \( S(r) = \frac{1}{r^4} \), which describes the resistance to blood flow as a function of the radius \( r \) of the blood vessel.

Step 2: Differentiate S(r) to Find the Rate of Change (Part a)

To find the rate of change of resistance with respect to \( r \), we need to differentiate \( S(r) \) with respect to \( r \). The differentiation is given by: \( S'(r) = \frac{d}{dr} \left( r^{-4} \right) \). Using the power rule, we find that: \( S'(r) = -4r^{-5} = -\frac{4}{r^5} \). This is the rate of change of resistance with respect to \( r \).

Step 3: Calculate Resistance at Given Radius (Part b)

Substitute \( r = 1.2 \) mm into the original function \( S(r) = \frac{1}{r^4} \): \( S(1.2) = \frac{1}{(1.2)^4} \). Calculating this yields: \( S(1.2) \approx 0.482 \). Thus, the resistance when the radius is 1.2 mm is approximately 0.482.

Step 4: Rate of Change at Specific Radius (Part c)

We need to find the rate of change of \( S \) with respect to \( r \) when \( r = 0.8 \) mm. Using \( S'(r) = -\frac{4}{r^5} \), substitute \( r = 0.8 \): \( S'(0.8) = -\frac{4}{(0.8)^5} \). Calculating this gives us: \( S'(0.8) \approx -9.77 \). Therefore, the rate of change of \( S \) at a radius of 0.8 mm is approximately -9.77.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that helps us understand and quantify how a function changes as its inputs change.

• It involves taking the derivative of a function.
• The derivative gives us the rate of change or the slope of the function at any given point.

In this exercise, we are dealing with the function \( S(r) = \frac{1}{r^4} \), where understanding the change in resistance with respect to changes in the radius \( r \) is crucial. To find how resistance changes, we differentiate \( S(r) \) using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Applying this rule, we find that \( S'(r) = -\frac{4}{r^5} \), which tells us how fast the resistance \( S \) decreases as the radius \( r \) increases. This negative exponent means as \( r \) gets larger, the value of the expression becomes significantly smaller, indicating a steep decrease in resistance.

Rate of Change

The rate of change is a concept used to describe how one quantity changes in relation to another. It is especially powerful for understanding how systems react to various influences.

• In calculus, the derivative of a function provides the rate of change.
• For the function \( S(r) = \frac{1}{r^4} \), the derivative \( S'(r) \) quantifies how resistance changes as radius changes.

When calculating the rate of change of resistance in a blood vessel, it helps to visualize how the attribute we measure, such as resistance in this case, reacts as the radius of the vessel changes. For example, at \( r = 0.8 \) mm, the rate of change \( S'(0.8) = -9.77 \) indicates that the resistance heavily decreases as the radius increases. Such a steep rate of change can have significant implications in biomedical contexts, especially as it relates to calculating parameters for medical devices such as stents. With a smaller radius, slight changes may lead to substantial variations in resistance.

Blood Flow Resistance

Blood flow resistance is a critical factor in understanding cardiovascular health. It refers to the opposition that the blood vessels provide against the flow of blood. The resistance determines how hard the heart has to work to pump blood through the circulatory system.

• Resistance is inversely related to the fourth power of the radius of the blood vessel.
• This means a small change in the radius can lead to a large change in resistance.

The formula \( S(r) = \frac{1}{r^4} \) captures the essence of how blood flow resistance behaves. A key insight from this formula is the exponential relationship: If the radius is halved, the resistance increases by a factor of 16, showing a highly sensitive response. Understanding this relationship helps healthcare professionals assess risks or design interventions, such as medication or surgery, to influence blood flow resistance, thereby maintaining a healthy circulatory system. Thus, accurately calculating the resistance at a specific radius, like \( r = 1.2 \) mm, provides essential information for diagnosis and treatment.