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Chapter 1: Problem 60

The number \(N\) of bacteria in a refrigerated food is given by $$N(T)=10 T^{2}-20 T+600, \quad 2 \leq T \leq 20$$ where \(T\) is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by $$T(t)=3 t+2, \quad 0 \leq t \leq 6$$ where \(t\) is the time in hours. (a) Find the composition \((N \circ T)(t)\) and interpret its meaning in context. (b) Find the bacteria count after 0.5 hour. (c) Find the time when the bacteria count reaches 1500 .

Short Answer

Expert verified
The composition of functions \(N(T)\) and \(T(t)\) is \(N(T(t)) = 10(3t+2)^2 - 20(3t+2) + 600\). The bacteria count after 0.5 hour is obtained by substituting \(t = 0.5\) into the composition function, which yields a bacteria count of 830. The time when the bacteria count reaches 1500 is approximately 1.85 hours.

Step by step solution

01

Finding the composition of functions \(N(T)\) and \(T(t)\)

Start by finding the composition of \(N\) and \(T\), which is written as \(N(T(t))\). To do this, substitute the function \(T(t) = 3t+2\) into the function \(N(T) = 10T^2 - 20T + 600\). The resulting function will represent the number of bacteria as a function of time.
02

Calculate the bacteria count after 0.5 hour

To find the bacteria count after 0.5 hour, substitute \(t = 0.5\) into the composition function \(N(T(t))\). Solve the equation to get the number of bacteria at this time.
03

Find the time when the bacteria count reaches 1500

To find the time when the bacteria count reaches 1500, set the composition function \(N(T(t)) = 1500\). Solve the equation to find the value of \(t\) at which the bacteria count is 1500. Ensure that the found value is within the range \(0 \leq t \leq 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
Function evaluation is the process of determining the output of a function for a given input. Think of it as a machine, where you input a number and it calculates an output based on its rule. This is a fundamental concept in mathematics, often used to connect real-world scenarios with mathematical models.
In our exercise, we have two functions: \(N(T) = 10T^2 - 20T + 600\) and \(T(t) = 3t + 2\). To find how bacterial count changes over time, we need to evaluate these functions.
  • Evaluate \(T(t)\): This tells us the temperature of food over time. Substitute different values of \(t\) to see how \(T\) changes, like setting \(t = 0.5\) to find temperature after half an hour.
  • Compose and evaluate \(N(T(t))\): The composition \((N \circ T)(t)\) plugs the result of \(T(t)\) into \(N(T)\). Each output of \(T(t)\) becomes the input for \(N(T)\).

This step ties temperature changes to the number of bacteria, linking our mathematical model to real-world conditions.
Quadratic Functions
Quadratic functions are pivotal in the study of algebra. These are polynomials of degree two and have the general form \(ax^2 + bx + c\). Their graphs are called parabolas, which can open upward or downward depending on the coefficient of the squared term.
In our problem, the function \(N(T) = 10T^2 - 20T + 600\) is quadratic. Here's what to note:
  • The coefficient 10 tells us the parabola opens upwards.
  • 'Vertex form' helps determine the max/min points, but here it's more straightforward to evaluate through function composition.
  • Every computed \(T\) value from \(T(t)\) is plugged back to \(N\), thus allowing us to understand how variations in \(T\) impact \(N\).

Understanding quadratics is key in finding critical points, such as maximum bacteria levels at specific temperatures, showing the practical utility of these mathematical forms.
Solving Equations
Solving equations is a crucial mathematical process that allows us to find unknown values that satisfy certain conditions. In the context of this exercise, equations help us determine when the bacteria count reaches specific levels.
Here's how to solve the equation \(N(T(t)) = 1500\):
  • Firstly, substitute \(T(t)\) into \(N(T)\) to establish a single equation dependent on \(t\).
  • Next, set \((N \circ T)(t) = 1500\) and solve for \(t\).
  • This involves manipulating the equation, potentially using methods like factoring or using the quadratic formula, given the quadratic nature of \((N \circ T)(t)\).

Finally, it's important to verify that the solution for \(t\) lies within the interval \(0 \leq t \leq 6\), to ensure it's valid within the context of the problem. This process exemplifies how equations serve as powerful tools to uncover meaningful solutions.

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Most popular questions from this chapter

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(v\) varies jointly as \(p\) and \(q\) and inversely as the square of s. \((v=1.5 \text { when } p=4.1, q=6.3, \text { and } s=1.2 .)\)

Assume that \(y\) is directly proportional to \(x .\) Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x .\) $$x=5, y=1$$

Use the fact that 13 inches is approximately the same length as 33 centimeters to find a mathematical model that relates centimeters \(y\) to inches \(x .\) Then use the model to find the numbers of centimeters in 10 inches and 20 inches.

An oceanographer took readings of the water temperatures \(C\) (in degrees Celsius) at several depths \(d\) (in meters). The data collected are shown as ordered pairs \((d, C)\). $$\begin{aligned} &(1000,4.2) \quad(4000,1.2)\\\ &(2000,1.9) \quad(5000,0.9)\\\ &(3000,1.4) \end{aligned}$$ (a) Sketch a scatter plot of the data. (b) Does it appear that the data can be modeled by the inverse variation model \(C=k / d ?\) If so, find \(k\) for each pair of coordinates. (c) Determine the mean value of \(k\) from part (b) to find the inverse variation model \(C=k / d\). (d) Use a graphing utility to plot the data points and the inverse model from part (c). (e) Use the model to approximate the depth at which the water temperature is \(3^{\circ} \mathrm{C}\).

Assume that \(y\) is directly proportional to \(x .\) Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x .\) $$x=5, y=12$$
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