Problem 58 - College Algebra

Question

An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, \(P,\) who contracted the flu \(t\) days after it broke out is given by the model \(P=-t^{2}+13 t+130,\) where \(1 \leq t \leq 6\) . Find the day that 160 students had the flu. Recall that the restriction on \(t\) is at most \(6 .\)

Step-by-Step Solution

Answer

160 students had the flu on day 3.

Step 1: Understand the Equation

Given the equation for the number of students who contracted the flu, \( P = -t^2 + 13t + 130 \), we need to find the value of \( t \) such that \( P = 160 \).

Step 2: Set Up the Equation

Substitute 160 for \( P \) in the equation to find \( t \): \(-t^2 + 13t + 130 = 160\).

Step 3: Rearrange to Standard Form

Bring all terms to one side of the equation to set it to zero: \(-t^2 + 13t + 130 -160 = 0\). Simplify to get \(-t^2 + 13t - 30 = 0\).

Step 4: Solve the Quadratic Equation

This is a quadratic equation in the form \(-t^2 + 13t - 30 = 0\). We use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -1 \), \( b = 13 \), and \( c = -30 \).

Step 5: Calculate the Discriminant

Calculate \( b^2 - 4ac \): \( 13^2 - 4(-1)(-30) = 169 - 120 = 49 \).

Step 6: Apply Quadratic Formula

Substitute \( a \), \( b \), and the discriminant into the quadratic formula: \( t = \frac{-13 \pm \sqrt{49}}{-2} \).

Step 7: Solve for t

Since \( \sqrt{49} = 7 \), substitute to find \( t = \frac{-13 + 7}{-2} \) and \( t = \frac{-13 - 7}{-2} \). This gives \( t_1 = 3 \) and \( t_2 = 10 \).

Step 8: Validate the Solution with Given Range

Since \( 1 \leq t \leq 6 \), choose \( t = 3 \) because \( t = 10 \) is outside of the given range.

Key Concepts

Epidemiological Modeling

Understanding epidemiological modeling involves analyzing how diseases spread within a population. In this exercise, the model used is a quadratic equation: \( P = -t^2 + 13t + 130 \). This equation reflects the number of students contracting the flu each day, where \( P \) is the number of students and \( t \) is the time in days after the outbreak.
Models like this are crucial in predicting the course of an epidemic and planning interventions.
They allow researchers to simulate different scenarios based on variables like infection rates and recovery times. By using such models, public health officials can make informed decisions to control or prevent outbreaks, minimizing the impact on communities.

Solving Quadratic Equations

Solving quadratic equations is an essential skill in algebra that involves finding the values of the variable that make the equation true. In this problem, we begin with the equation from the model: \(-t^2 + 13t - 30 = 0\). Quadratic equations typically take the form \( ax^2 + bx + c = 0 \), and our goal is to determine the values of \( t \) that satisfy this equation.

• First, set up the equation to equal zero, as demonstrated in the step-by-step solution. This standard form makes it easier to apply further methods of solving.
• Next, use methods such as factoring, completing the square, or the quadratic formula to find the roots of the equation. These roots represent potential solutions for the variable in question.

Understanding how to solve quadratics provides the foundation for more complex problem-solving scenarios.

Using Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of quadratic equations. For any equation of the form \( ax^2 + bx + c = 0 \), the formula is \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
In our example, we have the coefficients \( a = -1 \), \( b = 13 \), and \( c = -30 \).

• First, calculate the discriminant, \( b^2 - 4ac \). In this case, it's \( 49 \), revealing the nature of the solutions.
• Then, substitute the values into the formula to find the two potential solutions.

This process yields \( t = 3 \) and \( t = 10 \), but only \( t = 3 \) is valid due to the given range. The formula simplifies and ensures accuracy in solving quadratics, especially when factoring is complex.

Algebraic Problem Solving

Algebraic problem solving extends beyond simply finding a solution. It's about unraveling the details of a problem to understand and tackle it methodically. This exercise involves several critical steps:

• Comprehending the problem: Recognizing the underlying model and what it represents is essential.
• Setting up the equation: Translate word problems into mathematical expressions.
• Simplifying and solving: Use algebraic methods to simplify equations and solve for the unknown variables.
• Validating solutions: Check if the solutions make sense in the context of the problem. Here, only \( t = 3 \) is valid as it falls within \([1, 6]\).

This comprehensive approach not only helps in solving the current question but builds skills for tackling more complex equations in the future.