Question

MODELING WITH MATHEMATICS If you know the speed of sound waves \(v\) (in meters per second) in air, you can approximate the air temperature \(K\) (in kelvin) by using the equation $$ K(v)=\frac{v^2}{402.3} . $$ The function \(C(v)=K(v)-273.15\) approximates the air temperature (in degrees Celsius) when sound waves travel \(v\) meters per second. Write a rule for \(C\). What is the air temperature (in degrees Celsius) when sound waves travel 350 meters per second?

Short answer

After substituting the speed \(v = 350\) m/s into the derived rule for the function \(C(v)\), we find the air temperature to be \[C(350) ≈ {-1.35}\] degrees Celsius.

Step-by-step solution

Derivation of function C(v)

As the definition indicates, \(C(v) = K(v) - 273.15\). As the function of \(K(v)\) has already been given as \(K(v) = v^2/402.3\), we substitute it into \(C(v)\) and get: \(C(v) = v^2/402.3 - 273.15\).

Calculate C(350)

Now that we have the rule for \(C(v)\), we can substitute \(v = 350\) into the rule to calculate the air temperature in Celsius. Hence: \(C(350) = (350^2)/402.3 - 273.15\)

Calculation result

After performing the operations from Step 2, we arrive at the numerical result.

Key concepts

Mathematical Modeling

Mathematical modeling is the art of translating real-world scenarios into the language of mathematics, using equations, functions, or algorithms. To transform an abstract idea like the relationship between temperature and the speed of sound into an actionable formula, scientists and mathematicians create models based on observed data and theoretical principles. The equation provided in the exercise, \( K(v)=\frac{v^2}{402.3} \), exemplifies such a model; it embodies the underlying physics of sound propagation in air as a function of temperature.

The process begins by identifying the variables of interest — in this case, the speed of sound (\(v\)) and temperature (\(K\) in Kelvin). Through experimentation and data analysis, a correlation between these variables is established. This correlation is then expressed mathematically, often involving constant terms that have been finely tuned to fit experimental observations — like the constant 402.3 in our equation. When we use the model to calculate real-world quantities, such as the atmospheric temperature based on sound speed, we employ substitution and simplification to arrive at our desired quantity. This mathematical model then serves as a predictive tool that helps us understand how changes in one variable might affect the other.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). These functions graph as parabolas, which can open upwards or downwards depending on the sign of \(a\).

In the given exercise, the formula \( K(v) = \frac{v^2}{402.3} \) represents a quadratic function where the term \(v^2\) indicates that the temperature increases as the square of the speed of sound. This is a characteristic feature of quadratic behavior, reflecting that some physical relationships involve squared quantities. The quadratic function produces a parabolic graph, signifying that for every positive and negative value of the sound speed, there is a corresponding positive temperature value, which would only make sense physically in the positive domain.

Understanding the shape and nature of the quadratic function is crucial for interpreting the model's predictions. For example, we can determine that at zero speed, the temperature will not be absolute zero due to subtracting the constant term when converting to Celsius, which is important for solving real-world problems using this model.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact and understandable form. It expresses numbers as a multiplier between 1 and 10, and a power of 10. For instance, \(4.023 \times 10^2\) represents 402.3. This notation is particularly useful in the sciences and engineering where extreme values are common.

In our exercise, the constant 402.3 could be expressed in scientific notation as \(4.023 \times 10^2\). While this constant does not require scientific notation for simplicity, we often encounter constants and coefficients in physics and chemistry that are much more practical to express in this form, especially when dealing with squared or cubed terms, which can quickly escalate into very large or minuscule numbers. Proper use of scientific notation makes calculation more manageable and reduces the chances of error when multiplying or dividing numbers that span multiple orders of magnitude.

When students work with equations derived from scientific principles, the use of scientific notation can simplify the arithmetic involved and help them discern the magnitude of the numbers they are working with, fostering a better grasp of the scale and implications of the problem at hand.