Problem 54 - Brief Applied Calculus

Question

BIOMEDICAL: Blood Flow Nitroglycerin is often prescribed to enlarge blood vessels that have become too constricted. If the cross-sectional area of a blood vessel \(t\) hours after nitroglycerin is administered is \(A(t)=0.01 t^{2}\) square centimeters (for \(1 \leq t \leq 5)\), find the instantaneous rate of change of the cross-sectional area 4 hours after the administration of nitroglycerin.

Step-by-Step Solution

Answer

The instantaneous rate of change is 0.08 square centimeters per hour.

Step 1: Understand the Problem

We are given a function \(A(t) = 0.01t^2\) that represents the cross-sectional area of a blood vessel after nitroglycerin is administered. We need to find the instantaneous rate of change of \(A(t)\) at \(t = 4\) hours, which means we are looking for the derivative of \(A(t)\) at \(t = 4\).

Step 2: Find the Derivative

The instantaneous rate of change at a point is given by the derivative of the function at that point. Thus, we need to find \(A'(t)\). Start by finding the derivative of the given function \(A(t) = 0.01t^2\): \[ A'(t) = \frac{d}{dt}(0.01t^2) = 0.01 \times 2t = 0.02t. \]

Step 3: Evaluate the Derivative at \(t = 4\)

Substitute \(t = 4\) into the derivative \(A'(t)\) to find the instantaneous rate of change of the area at \(t = 4\):\[ A'(4) = 0.02 \times 4 = 0.08. \] This means that the rate of change of the cross-sectional area at \(t = 4\) hours is 0.08 square centimeters per hour.

Key Concepts

Instantaneous Rate of Change

The instantaneous rate of change is a crucial concept in calculus that helps us understand how fast something is changing at a specific moment in time. To find the instantaneous rate of change, we look at the derivative of a function at a particular point. For the problem we're considering, the mathematical expression described how the cross-sectional area of a blood vessel changes over time due to the administration of nitroglycerin. At exactly 4 hours after the nitroglycerin begins to work, we want to know how quickly the area of the blood vessel is increasing. This change tells us potentially how effectively the drug is working. To find this rate, we calculate the derivative of the function that represents the blood vessel's area. By doing this, we determine how the area is changing instantaneously, giving us insight into the process or effectiveness at that specific moment, 4 hours in this case.

Derivatives

Derivatives are one of the most fundamental concepts in calculus and represent a tool that allows us to examine the rate at which a quantity changes. It is essentially the "speed" of the change of a function. In the original problem, our function is the area of a blood vessel, represented as \(A(t) = 0.01t^2\). To find out how this area changes instantaneously, we need to find its derivative. The derivative of a function is typically calculated using the rules of differentiation. For the function \(A(t) = 0.01t^2\), using the power rule of derivatives, we find:

• The derivative of \(t^2\) is \(2t\).
• Multiplying by the constant 0.01 gives us \(A'(t) = 0.02t\).

The expression \(A'(t) = 0.02t\) is the derivative of the original function, which can be evaluated at any time \(t\) to find the instantaneous rate of change at that specific time.

Biomedical Applications

In biomedical applications, calculus and specifically the concept of derivatives play an important role in understanding and modeling biological processes. The problem we discussed involved how nitroglycerin affects the blood vessels, measured through the change in their cross-sectional area over time. This kind of modeling is vital in the medical field because it allows researchers and medical professionals to predict the behavior of biological systems under treatment. By knowing the rate at which blood vessels expand, doctors can adjust dosage or treatment plans to ensure optimal outcomes for patients. In practice, derivatives enable biologists and medical scientists to quantify how quickly drugs take effect, how quickly they are metabolized, and how different variables, like time, affect biological systems. This powerful tool helps in optimizing treatments and understanding diseases, making it an essential component of modern biomedical science.