Question

A researcher is studying the effects of caffeine on the body. As part of her research, she monitors the levels of caffeine in a person's bloodstream over time after drinking coffee. The function \(C(t)=\frac{50 t}{1.2 t^{2}+5}\) models the level of caffeine in one particular person's bloodstream, where \(t\) is the time, in hours, since drinking the coffee and \(C(t)\) is the person's bloodstream concentration of caffeine, in milligrams per litre. How long after drinking coffee has the person's level dropped to \(2 \mathrm{mg} / \mathrm{L} ?\)

Short answer

0.20 hours

Step-by-step solution

Identify the Given Information

The given function is the caffeine concentration in the bloodstream: \[C(t) = \frac{50t}{1.2t^2 + 5}\] We need to find the time t when the concentration drops to 2 mg/L, i.e., \[C(t) = 2\]

Set Up the Equation

Set the function equal to 2 and solve for t: \[\frac{50t}{1.2t^2 + 5} = 2\]

Clear the Denominator

Multiply both sides by \(1.2t^2 + 5\) to eliminate the fraction: \[50t = 2(1.2t^2 + 5)\]

Simplify the Equation

Expand and simplify the equation: \[50t = 2.4t^2 + 10\]Rearrange the equation to form a standard quadratic equation: \[2.4t^2 - 50t + 10 = 0\]

Solve the Quadratic Equation

Use the quadratic formula \[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Identify the coefficients: \(a = 2.4\), \(b = -50\), and \(c = 10\)Then the solutions for t are: \[t = \frac{50 \pm \sqrt{(-50)^2 - 4(2.4)(10)}}{2(2.4)}\]Calculate the discriminant: \(2500 - 96 = 2404\)So: \[t = \frac{50 \pm \sqrt{2404}}{4.8}\]Approximate the solutions: \[t = \frac{50 \pm 49.04}{4.8}\]\(t_1 \approx 20.66\) and \(t_2 \approx 0.20\)Since time must be positive: \(t \approx 0.20\)

Key concepts

Quadratic Equation

The problem involves solving a quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\).Quadratic equations frequently arise in various contexts, including physics, engineering, and even finance. To solve these equations, we often use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].In this case, the quadratic equation we derived from the caffeine concentration model is \[2.4t^2 - 50t + 10 = 0\],where \(a = 2.4\), \(b = -50\), and \(c = 10\).By substituting these values into the quadratic formula, we can find the possible values of \(t\).In this exercise, we calculated the solutions to be approximately \(t_1 \approx 20.66\) and\(t_2 \approx 0.20\).Since time must be positive, \(t \approx 0.20\).

Caffeine Metabolism

Caffeine metabolism refers to how the body processes and eliminates caffeine, a stimulant found in coffee and other beverages. The concentration of caffeine in the bloodstream decreases over time as the body metabolizes it. Factors such as age, gender, and genetic makeup can affect how quickly an individual metabolizes caffeine. In this exercise, we are given a specific function, \[C(t)=\frac{50t}{1.2t^2+5}\],which models the decline in caffeine concentration in a person's bloodstream over time, with \(t\) representing the time elapsed since drinking coffee. By solving this function, we can find out how long it takes for the caffeine concentration to drop to a certain level, in this case, \(2\) mg/L.

Mathematical Modelling

Mathematical modelling involves using mathematical equations and functions to represent real-world phenomena. In this exercise, we use a mathematical model, \[C(t)=\frac{50t}{1.2t^2+5}\], to represent the concentration of caffeine in a person's bloodstream as a function of time. This allows researchers to predict and study the effects of caffeine over a period. Mathematical models are valuable tools in many fields, including biology, economics, and engineering, as they help simulate scenarios, test hypotheses, and make predictions. By manipulating and solving the provided function, we can understand and investigate the behavior of caffeine in the bloodstream.

Solving for variable in function

In this exercise, we needed to find the time,\(t\), at which the caffeine concentration reaches \(2\) mg/L. This involves setting the function equal to \(2\) and solving for \(t\). Here's a quick look at the process we followed:

• Set \(C(t) = \frac{50t}{1.2t^2 + 5} = 2\)
• Multiply both sides by \(1.2t^2 + 5\) to clear the fraction:\(50t = 2(1.2t^2 + 5)\)

• Simplify the equation to form a standard quadratic equation: \(2.4t^2 - 50t + 10 = 0\)
• Use the quadratic formula to solve for \(t\): \[t = \frac{50 \pm \sqrt{2404}}{4.8}\]
• Approximate the solutions and choose the positive one: \(t \approx 0.20\)

This involved several algebraic steps and the application of the quadratic formula, which is a fundamental technique in algebra that helps in solving quadratic equations.