Question

The number of bacteria in a certain food product is given by \(N(T)=10 T^{2}-20 T+600, \quad 1 \leq T \leq 20\) where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature of the food is given by \(T(t)=3 t+1\) where \(t\) is the time in hours. Find (a) the composite function \(N(T(t))\) and (b) the time when the bacteria count reaches 1500 .

Short answer

The composite function is \(N(T(t)) = 90t^2 + 260t + 590\). The bacteria count reaches 1500 at \(t = 0.911\) hours.

Step-by-step solution

Finding the Composite Function

To find the composite function \(N(T(t))\), we need to replace \(T\) in \(N(T)\) function with the \(T(t)\) function.\nSo, \(N(T(t)) = N(3t+1) = 10*(3t+1)^2 - 20*(3t+1) + 600.

Simplify the Composite Function

Simplify \(N(T(t))\) by expanding and simplifying it. This should yield \( N(T(t))= 90t^2 + 260t + 590.\)

Find the time when bacteria count reaches 1500

Set \(N(T(t)) = 1500\) and solve for \(t\): \n1500 = 90t^2 + 260t + 590 \nThis is a quadratic equation in the form \(at^2 + bt + c = 0\), which can be solved using the quadratic formula \(-b \pm \sqrt{b^2 - 4ac}\) / \(2a\).

Calculate the Time

Applying the quadratic formula to our equation, we find that \(t = 0.911\) hours.

Key concepts

Bacteria Growth Model

Understanding the bacteria growth model is crucial in various fields, such as food safety, healthcare, and environmental biology. In our exercise, the growth of bacteria is modeled as a function of temperature: the higher the temperature, the faster bacteria are likely to grow. The function given,
\(N(T)=10 T^{2}-20 T+600\), represents how the number of bacteria, denoted by \(N\), changes with temperature \(T\). Temperature plays a significant role here because certain ranges can either inhibit or encourage bacterial proliferation.
A critical aspect of tackling problems like these in mathematical biology is to connect the variables involved - in this case, temperature and time - to see how changes over time affect the growth. A composite function expresses one variable in terms of another, allowing us to explore this relationship. When we say \(N(T(t))\), we are essentially looking to understand how the number of bacteria evolves over time as temperature changes accordingly. This real-world application demonstrates how mathematical models can effectively predict changes in biological systems.

Quadratic Equations

At the heart of this exercise lies a quadratic equation, a fundamental component of algebra. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable.
In our example, the equation to find the time when bacteria count reaches 1500 is represented as \(90t^2 + 260t + 590 = 1500\). This equation is quadratic in nature because of the term \(90t^2\), which has the variable raised to the second power, coding for a curved graph when plotted.

Why Quadratic Equations Matter

Quadratic equations are pivotal in various disciplines, from physics to finance, because they describe many natural processes involving acceleration, area calculations, and profit maximization. They offer insights into points of maximum and minimum, rates of change, and the nature of physical systems in equilibrium.
In this scenario, solving the quadratic equation enables us to predict the time at which the bacteria count will hit a specific threshold - a vital piece of information for ensuring food safety.

Algebraic Manipulation

Algebraic manipulation is an indispensable tool in simplifying and solving mathematical expressions and equations. It involves various techniques such as expanding brackets, collecting like terms, and isolating variables to find solutions to different kinds of equations.
In the exercise, we performed algebraic manipulation by first finding the composite function and then expanding it to simplify the expression. This process involved squaring the binomial \((3t+1)^2\), distributing the constant 10 across the resulting terms, and further simplifying to obtain \(90t^2 + 260t + 590\).

The Importance of Clear Steps

Providing step-by-step solutions facilitates understanding the chain of reasoning that leads from the initial problem to the final answer. It's not just about the answer but about how we reach it that enriches students' learning experience. We simplify complex expressions to a form that is easier to work with or to make patterns more evident, strengthening our algebraic intuition and problem-solving skills. Thus, algebraic manipulation isn’t just a process; it’s a fundamental aspect of mathematical literacy.