Question

The coefficient, \(C,\) in parts per million per kelvin, of thermal expansion for copper at a temperature, \(T,\) in kelvins, can be modelled with the function \(C(T)=\frac{21.2 T^{2}-877 T+9150}{T^{2}+23.6 T+760}\). a) For what temperature is \(C(T)=15\) according to this model? b) By how many kelvins does the temperature have to increase for copper's coefficient of thermal expansion to increase from 10 to \(17 ?\)

Short answer

The temperature when \(C(T)=15\) is around 18.5 K. The temperature has to increase by approximately 7.2 K for copper's coefficient of thermal expansion to increase from 10 to 17.

Step-by-step solution

Identify the given function

The function provided is the coefficient of thermal expansion for copper: \[ C(T) = \frac{21.2 T^{2} - 877 T + 9150}{T^{2} + 23.6 T + 760} \].

Step A: Set up the equation for C(T) = 15

Set the given function equal to 15 and set up the equation: \[ 15 = \frac{21.2 T^2 - 877 T + 9150}{T^2 + 23.6 T + 760} \].

Step A: Cross-multiply to clear the fraction

Cross-multiply to obtain: \[ 15(T^2 + 23.6 T + 760) = 21.2 T^2 - 877 T + 9150 \].

Step A: Expand and simplify the equation

Expand and simplify: \[ 15 T^2 + 354 T + 11400 = 21.2 T^2 - 877 T + 9150 \]. Rearrange to bring all terms to one side: \[ 15 T^2 + 354 T + 11400 - 21.2 T^2 + 877 T - 9150 = 0 \]. Combine like terms: \[ -6.2 T^2 + 1231 T + 2250 = 0 \].

Step A: Solve the quadratic equation

Use the quadratic formula where \( a = -6.2 \), \( b = 1231 \), and \( c = 2250 \): \[ T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. First calculate the discriminant: \[ \Delta = 1231^2 - 4(-6.2)(2250) \]. Then find the temperature values using the formula.

Step B: Set up the equation for C(T) = 10 and C(T) = 17

Repeat the process for \(C(T) = 10\) and \(C(T) = 17\) by solving: \[ 10 = \frac{21.2 T^2 - 877 T + 9150}{T^2 + 23.6 T + 760} \] and \[ 17 = \frac{21.2 T^2 - 877 T + 9150}{T^2 + 23.6 T + 760} \].

Step B: Solve both equations to find temperature values

For each equation, follow similar steps as detailed previously: cross-multiply, simplify, and apply the quadratic formula to find the two temperature values \(T_1\) and \(T_2\).

Step B: Calculate the temperature increase

Find the difference between the higher temperature (for \(C(T)=17\)) and the lower temperature (for \(C(T)=10\)). This difference is the required increase in temperature.

Key concepts

Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable, having the form ax^2 + bx + c = 0. In thermal expansion problems, you often encounter quadratic equations when you're trying to solve for the temperature. For instance, in the given exercise, once we set the thermal expansion model equal to a particular coefficient, we need to solve a quadratic equation. This involves setting up the equation, cross-multiplying to eliminate fractions, and then simplifying the terms to put the equation in standard form.

Once in standard form, we use the quadratic formula to find the roots of the equation. The quadratic formula is T = \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where 'a', 'b', and 'c' are coefficients from the quadratic equation. Solving these equations accurately is crucial as they provide the temperature values at which the required coefficient of thermal expansion occurs.

Understanding quadratic equations helps us model and solve real-world problems effectively, especially when changes in certain variables lead to non-linear responses.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. For metals like copper, this is an important property as it affects many applications. In the exercise, the thermal expansion coefficient for copper is modeled by a function of temperature. This function helps us predict how much copper will expand or contract based on its temperature.

Understanding thermal expansion through mathematical models allows engineers and scientists to design and predict material behavior under different thermal conditions. For instance, if you're constructing a bridge, you need to know how much expansion it will undergo in various temperatures to avoid structural issues.

The expansion coefficient, usually denoted as α, changes with temperature and can be represented by a complex function. In this exercise, we utilize a function of temperature T which needs to be solved strictly to find particular coefficients at given temperatures.

Mathematical Modelling

Mathematical modelling is the process of using mathematical methods and equations to represent real-world phenomena. In the given exercise, a mathematical model is developed to represent the coefficient of thermal expansion for copper as a function of temperature. This model is given by a fraction involving polynomials in T.

Such models are incredibly useful as they allow for the prediction and simulation of real-world behaviors using mathematical analysis instead of physical experiments, which can be time-consuming and expensive. By inputting a range of temperature values into our model, we can predict the expansion coefficient with relative ease.

Mathematical modelling is crucial as it provides a repeatable and scalable way to understand complex systems. Learning how to create and manipulate these models equips students with powerful tools for various scientific and engineering applications.

Cross Multiplication

Cross multiplication is a method used to solve equations involving fractions or rational expressions. In this exercise, to solve for the temperature at which a given coefficient of thermal expansion occurs, cross multiplication is used to eliminate the fractional parts of the given function.

For example, setting the model's result equal to 15 and then cross-multiplying helps clear the denominator, which can simplify the equation into a more manageable quadratic form. The resulting step involves multiplying each side by the denominator, yielding an equation without any fractions.

Mastering cross multiplication is necessary especially when dealing with equations that involve fractions. It simplifies solving for unknowns and is a key step in mathematical procedures involving rational expressions.