Problem 61 - College Algebra and Calculus: An Applied Approach

Question

The number of bacteria in a certain food product is given by \(N(T)=25 T^{2}-50 T+300, \quad 2 \leq T \leq 20\) where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature of the food is given by \(T(t)=2 t+1\) where \(t\) is the time in hours. Find (a) the composite function \(N(T(t))\) and \((\mathrm{b})\) the time when the bacteria count reaches 750 .

Step-by-Step Solution

Answer

a) The composite function \(N(T(t)) = 100t^{2} + 100t + 175\). b) The time at which the bacteria count reaches 750 can be determined using the quadratic formula. Discard any negative time solution obtained.

Step 1: Find the Composite Function

We substitute \(T(t)\) into the function \(N(T)\) to get the composite function \(N(T(t))\). So, wherever we see \(T\) in the function \(N(T)\), we replace it with \(2t+1\). Doing this yields: \(N(T(t)) = N(2t + 1) = 25(2t+1)^{2} -50(2t+1) + 300\). This is our composite function which will give us the number of bacteria as a function of time.

Step 2: Simplify the Composite Function

We can simplify this expression by expanding the brackets and combining like terms. Doing this gives us: \(N(T(t)) = 100t^{2} + 100t + 175\). This is a simplified version of our composite function.

Step 3: Equate Composite Function to 750 and Solve for t

To solve for when the bacteria count is 750, we set our composite function equal to 750 and solve for \(t\). This gives us the equation \(100t^{2} + 100t - 575 = 0\). The solutions for \(t\) can be calculated by using the quadratic formula \(-b ± \sqrt{b^{2}-4ac} \over 2a\). The solutions will yield two possible times, however, since time cannot be negative we discard the negative solution.

Key Concepts

Quadratic Functions

Quadratic functions are an essential part of understanding how various phenomena, such as bacteria growth in this context, change with respect to other variables. A quadratic function is generally in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In this exercise, the number of bacteria as a function of temperature is represented by a quadratic function: \[N(T) = 25T^2 - 50T + 300\].This equation tells us that as temperature \(T\) changes, the bacterial count will follow a parabolic path.

• The coefficient \(25\) represents how quickly the bacteria number changes with the square of the temperature.
• The linear term \(-50T\) adjusts the shape and direction of the parabola.
• The constant \(300\) shifts the graph upwards by 300 units, making it the initial bacteria count when \(T = 0\).

Understanding these aspects of quadratic functions helps when combining them with other functions, like in composite functions.

Bacteria Growth

Bacteria growth depends on several factors, including temperature. In this exercise, the bacteria count changes with temperature in a specific way, described by the function \(N(T)\). The function forms a parabola due to its quadratic nature, which means the number of bacteria doesn't grow linearly. Instead, the growth rate itself changes, accelerating or decelerating as temperature changes. The composite function \(N(T(t))\) simplifies understanding how bacteria count varies over time, making it essential for situations involving changing environmental conditions.

• Initially, simplification of \(N(T(t))\) to \(100t^2 + 100t + 175\) helps analyze bacterial growth over time \(t\).
• It allows prediction of the bacteria count without directly measuring temperature every moment.
• This analysis becomes crucial when determining specific bacteria counts at different times, like at 750, to find suitable conditions.

Knowing how bacteria grow in response to changing temperatures aids in preventing spoilage in food management.

Temperature Impact on Bacteria

Temperature is a major influence on bacterial growth, impacting their multiplication rate and survival. In this problem, the relationship between temperature and bacteria count is modeled as a function of time, \(T(t) = 2t + 1\). This function maps time to temperature, indicating that as time progresses, temperature linearly increases.

• Each hour, temperature rises by 2 degrees, starting from 1 degree at \(t=0\).
• This increase accelerates bacterial growth as depicted by the quadratic function \(N(T)\).
• The intersection of mathematical modeling and biology showcases how vital temperature control is in inhibiting unwanted bacterial multiplication.

Analyzing the function helps us understand the breaking point where bacteria reach unsafe levels, which is essential in food safety and refrigeration processes. By studying \(N(T(t))\), predictions can be made to determine how long food can stay safely at a given temperature before bacterial count becomes a health concern.