Problem 98 - College Algebra

Question

The common cold is caused by a rhinovirus. The polynomial $$0.75 x^{4}+3 x^{3}+5$$ describes the billions of viral particles in our bodies after \(x\) days of invasion. Use a calculator to find the number of viral particles after 0 days (the time of the cold's onset), 1 day, 2 days, 3 days, and 4 days. After how many days is the number of viral particles at a maximum and consequently the day we feel the sickest? By when should we feel completely better?

Step-by-Step Solution

Answer

The function evaluates to 5, 8.75, 19.5, 45.75, and 97 billion viral particles at days 0, 1, 2, 3, and 4, respectively. The virus peaks on day 3 before starting to alleviate on day 4.

Step 1: Evaluate the function at different days

Use the given function \(0.75 x^{4}+3 x^{3}+5\) to calculate the number of viral particles for each specified day \(x\). For instance, when \(x=0\), the number of viral particles is \(0.75 *0^{4}+3 *0^{3}+5 = 5\) billion.

Step 2: Find the maximum of the function

The maximum is the highest point of the function and occurs where the derivative is zero. Derive the function to get \(3x^{2}+9x\). Solve \(3x^{2}+9x=0\) for \(x\), yielding the values where the maximum, minimum, or flat points occur.

Step 3: Determine the point of maximum infection

Substitute each value from step 2 into the second derivative to identify whether the point is a maximum or minimum. If the second derivative is negative at that point, it's a maximum.

Step 4: Identify when the patient should recover

The patient should be better when the number of viral particles is at its minimum. This typically corresponds to the end of the given timeframe, which is day 4. Validate this by substituting \(x = 4\) into the original function.

Key Concepts

College Algebra

College algebra is a branch of mathematics that covers algebraic concepts often at a level that prepares students for future studies in fields such as engineering, science, economics, and more. It includes the study of functions, equations, inequalities, and polynomials, such as the one used to model the growth of viral particles in our example.

In our exercise, we observe how algebra is applied in a real-world scenario, providing a meaningful context to mathematical concepts. Understanding how to manipulate and evaluate polynomials, and how to find their extrema, which are common topics in college algebra, directly relate to solving problems about virus spread and recovery times.

Viral Particle Modeling

Viral particle modeling is an application of mathematical functions to represent how an infection progresses over time. In the given exercise, the polynomial function \[\begin{equation}0.75 x^{4}+3 x^{3}+5\tag{1}\text{billion viral particles}\tag{2}\text{is employed to predict the numbers of rhinovirus particles over the course of an infection.}\end{equation}\]

The use of a polynomial provides an idealized curve that can be analyzed to understand important aspects of the infection, such as the rate of increase of viral particles and when it is expected to peak. This is crucial for medical professionals and researchers who are trying to predict the course of an outbreak and the potential impact on individual patients and the wider population.

Function Evaluation

Function evaluation is a core operation in algebra that involves substituting a number into a function and calculating the output. This is pivotal in understanding the behavior of functions under various conditions. For instance, in the example, function evaluation helps us calculate the number of viral particles after 0, 1, 2, 3, and 4 days by substituting each of these values for x in the polynomial.

By systematically evaluating the polynomial at these different time points, we can trace the trend of the infection, providing tangible numbers to better understand and visualize the spread of the virus over the given period. This concept highlights the importance of algebra skills in real-world situations.

Local Extrema of Functions

A critical application of calculus in the realm of algebra is finding the local extrema of functions, which refer to the points where the function achieves local maximum or minimum values. These points provide crucial information about the behavior of the function, such as identifying the peak of infection in the case of our viral model.

In order to find the local extrema, as shown in the solution steps, we first derive the function to find where its slope is zero. These points are potential candidates for local maxima or minima. By applying the second derivative test, we can determine the nature of these points - whether they are points of maximum infection (local maxima) or potential recovery (local minima). Understanding these concepts allows students to analyze the progression of various phenomena, from biology to economics.