Problem 27 - Algebra 2 and Trigonometry

Question

The relationship between temperature and the rate at which crickets chirp can be approximated by the function \(\mathrm{n}(x)=4 x-160\) where \(\mathrm{n}\) is the number of chirps per minute and \(x\) is the temperature in degrees Fahrenheit. On a given summer day, the temperature outside between the hours of 6 A.M. and 12 P.M. can be modeled by the function \(f(x)=0.55 x^{2}+\) \(1.66 x+50\) where \(x\) is the number of hours elapsed from 6 P.M. a. Find the composite function n of. b. What is the rate of chirping at 11 A.M.?

Step-by-Step Solution

Answer

The rate of chirping at 11 AM is 128.2 chirps per minute.

Step 1: Define Composite Function

To find the composite function \(n(f(x))\), substitute the expression for \(f(x)\) into the function \(n(x)\). The function \(n(x) = 4x - 160\) becomes \(n(f(x)) = 4(f(x)) - 160\).

Step 2: Substitute f(x) into n(x)

Replace \(f(x)\) in \(n(f(x)) = 4(f(x)) - 160\) with \(0.55x^2 + 1.66x + 50\). This gives \(n(f(x)) = 4(0.55x^2 + 1.66x + 50) - 160\).

Step 3: Simplify the Expression for n(f(x))

Distribute the 4 inside the parentheses: \(n(f(x)) = 4 \times 0.55x^2 + 4 \times 1.66x + 4 \times 50 - 160\), simplifying to \(n(f(x)) = 2.2x^2 + 6.64x + 200 - 160\).

Step 4: Simplify Further

Combine the constant terms: \(n(f(x)) = 2.2x^2 + 6.64x + 40\). This is the composite function \(n(f(x))\).

Step 5: Determine the Number of Hours Since 6 AM for 11 AM

Since 11 AM is 5 hours after 6 AM, set \(x = 5\) in the composite function \(n(f(x))\).

Step 6: Evaluate Composite Function at x = 5

Substitute \(x = 5\) into \(n(f(x)) = 2.2x^2 + 6.64x + 40\), which gives \(n(f(5)) = 2.2(5)^2 + 6.64(5) + 40\).

Step 7: Calculate Chirping Rate

Evaluate the expression: \(n(f(5)) = 2.2(25) + 6.64(5) + 40\). Calculate these values: \(55 + 33.2 + 40\), which sums to \(128.2\). The rate of chirping at 11 AM is 128.2 chirps per minute.

Key Concepts

Quadratic Functions

Quadratic functions are a special type of polynomial function where the highest exponent of the variable is 2. They are commonly expressed in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratic functions graph as a parabola, which can open upwards or downwards depending on the sign of the coefficient \(a\). The vertex of the parabola, where it changes direction, is a key feature. In our exercise, the quadratic function \(f(x) = 0.55x^2 + 1.66x + 50\) models temperature changes over time.

• The coefficient \(a = 0.55\) tells us how "wide" or "narrow" the parabola is.
• The sign of \(a\) being positive indicates the parabola opens upwards.
• The coefficients \(b = 1.66\) and \(c = 50\) adjust the position of the vertex.

Understanding the shape and direction of a quadratic function helps in visualizing and predicting the behavior of the system it models, such as temperature fluctuations in this context.

Function Notation

Function notation is an essential part of mathematics that allows us to represent and work with functions in a simple and clear way. It typically uses symbols like \(f(x)\) to denote a function \(f\) with an input \(x\). This concise notation helps easily specify the operations to be performed on different inputs.
In our exercise, we encounter several functions:

• \(n(x) = 4x - 160\), which represents the relationship between the temperature and the chirping rate of crickets.
• \(f(x) = 0.55x^2 + 1.66x + 50\), modeling how temperature changes over time.

By using function notation, it's straightforward to manipulate and combine these functions into a composite function, which is crucial for solving more complex problems such as determining how chirping rates change with temperature. It also simplifies substituting values and solving for specific outputs as demonstrated in finding the chirping rate at a given time.

Rate of Change

The rate of change in a function describes how a function's output value changes as its input changes. It is a basic concept in calculus and helps us understand how fast or slow a function is changing. For linear functions like \(n(x) = 4x - 160\), the rate of change is constant, indicated by the coefficient of \(x\), in this case, 4.
When it comes to temperature-induced chirping, the function \(n(x)\) tells us the chirping rate increases by 4 chirps per minute for each degree of temperature increase. As for the composite function that combines temperature over time, \(n(f(x)) = 2.2x^2 + 6.64x + 40\), things are a bit more complex, due to the quadratic nature. Here, the rate of change is not constant and varies according to \(x\).

• This means the chirping rate accelerates or decelerates non-linearly based on the changes in temperature over time.
• The initial part of the time interval might see slower chirping rate changes compared to later times when changes accumulate more.

Understanding the rate of change helps in planning and interpreting real-world phenomena and their implications, such as predicting how environmental factors like temperature affect biological activity.

Polynomial Functions

Polynomial functions are mathematical expressions that can include multiple terms, each consisting of a variable raised to a power and multiplied by a coefficient. The degree of a polynomial is determined by the highest power of the variable. Quadratic functions, like \(f(x) = 0.55x^2 + 1.66x + 50\), are a subset of polynomial functions with a degree of 2.
Polynomial functions are very flexible and can model a wide variety of complex systems. They can have turned back after ups or downs known as bends or curves, like a hill and valley inside a rollercoaster ride, which makes them very useful in forecasting patterns.

• The function \(f(x) = 0.55x^2 + 1.66x + 50\) expresses how temperature varies over time, capturing changes that aren't purely linear or simply fixed.
• This type of function typically requires more thorough analysis to solve and understand due to the multiple terms involved.

The process of understanding polynomial functions involves examining how individual coefficients affect the graph and behavior of the entire function, providing insights into the storyline beyond simple cause-effect relationships.