Problem 97 - College Algebra with Corequisite Support

Question

The number of bacteria in a refrigerated food product is given by \(N(T)=23 T^{2}-56 T+1,3

Step-by-Step Solution

Answer

The composite function is \(N(T(t)) = 575t^2 + 65t - 31.25\), and the bacteria count reaches 6752 at \(t \approx 3.48\) hours.

Step 1: Substitute Temperature Function

To find the composite function \(N(T(t))\), start by substituting the temperature function \(T(t) = 5t + 1.5\) into the bacteria count function \(N(T) = 23T^2 - 56T + 1\).

Step 2: Expand and Simplify Composite Function

Substitute \(T(t)\) into \(N(T)\): \[N(5t + 1.5) = 23(5t + 1.5)^2 - 56(5t + 1.5) + 1\]. First, expand \((5t + 1.5)^2 = 25t^2 + 15t + 2.25\). Substitute this back:\[N(5t + 1.5) = 23(25t^2 + 15t + 2.25) - 56(5t + 1.5) + 1\].Expanding gives:\[575t^2 + 345t + 51.75 - 280t - 84 + 1\],which simplifies to \[575t^2 + 65t - 31.25\]. Thus, the composite function is \(N(T(t)) = 575t^2 + 65t - 31.25\).

Step 3: Set up the Equation for 6752 Bacteria Count

We need to find \(t\) when the bacteria count reaches 6752. Set the composite function equal to 6752:\[575t^2 + 65t - 31.25 = 6752\].

Step 4: Simplify the Equation

Rearrange the equation to standard quadratic form: \[575t^2 + 65t - 6783.25 = 0\].

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) where \(a = 575\), \(b = 65\), and \(c = -6783.25\). First compute \(b^2 - 4ac = 65^2 - 4 \times 575 \times -6783.25\). Then compute the square root and solve for \(t\).

Step 6: Evaluate and Round

Calculate using \(t = \frac{{-65 \pm \sqrt{{65^2 + 4 \times 575 \times 6783.25}}}}{{1150}}\). Evaluate to find two potential solutions for \(t\). Choose the solution within the valid range \(3.0 < 5t + 1.5 < 33\), which corresponds to \(0.3 < t < 6.3\). Round to two decimal places.

Key Concepts

Quadratic Equation

In mathematics, a quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \) where \( x \) represents an unknown, and \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). It is called "quadratic" because "quad" means "square," and the equation is about squaring what comes after it. Solving a quadratic equation typically involves a few different methods:

• Factoring: If the quadratic can be factored, it can be rewritten as a product of two binomials.
• Completing the Square: This method involves rearranging the quadratic to make it a perfect square trinomial.
• Quadratic Formula: The formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \) derives solutions for any quadratic equation.

In our problem, the quadratic comes into play where we find the necessary time \( t \) such that the bacteria population reaches a specific number. This requires rearranging a complex composite function back into standard quadratic form and utilizing the quadratic formula to find \( t \). Once obtained, solutions are evaluated within a defined range of validity.

Bacteria Population

In this context, the bacteria population is expressed as a function of temperature, indicating how the number of bacteria changes with a change in temperature. The function \( N(T) = 23T^2 - 56T + 1 \) captures this relationship by incorporating a quadratic equation, which provides a flexible model for describing biological growth phenomena.
To better understand this composite function, consider its two main components:

• Growth Rate: The quadratic expression reflects how rapidly the population increases with temperature, reaching a peak or diminishing depending on temperature values.
• Dynamic Range: The given range \( 3 < T < 33 \) helps maintain realism, ensuring the model predictions stay within biologically plausible scenarios.

By understanding this function, students prepare for analyzing more intricate biological data and simulating growth under varying environmental conditions, as seen in situations like bacterial contamination in food products. The quadratic relationship highlights the importance of temperature control in preserving perishable goods.

Temperature Function

The temperature function \( T(t) = 5t + 1.5 \) describes how the temperature of the food changes over time once it is removed from the refrigerator. Here, \( t \) is time in hours, and the linear form specifies a steady rate of temperature increase. This function plays a critical role as it enables us to understand how quickly the conditions for bacteria growth change over time.
Key points of understanding this function include:

• Initial Temperature: When \( t = 0 \), \( T(t) = 1.5 \). This indicates the starting temperature, likely just outside refrigeration.
• Rate of Change: The coefficient of \( t \) (which is 5 here) controls how fast the temperature is increasing per hour.

By integrating this temperature function into the bacteria population equation, we derive a composite function that dictates the bacteria count as a function of time rather than temperature. This transition from simple linear to composite quadratic functions is essential for modeling real-world scenarios where one variable depends on the change or variation of another over time.