Question

Use the following information. At sea level, the speed of sound in air is linearly related to the air temperature. If it is \(35^{\circ} \mathrm{C},\) sound will travel at a rate of 352 meters per second. If it is \(15^{\circ} \mathrm{C}\) sound will travel at a rate of 340 meters per second. If sound travels at a rate of 346 meters per second at sea level, what is the temperature?

Short answer

The temperature at which sound would travel at a rate of 346 meters per second at sea level is \(25^{\circ}C.\)

Step-by-step solution

Calculating the Slope of the Line (m)

Firstly, calculate the slope of the line, which represents the change in speed of sound per unit change in temperature. The slope \(m\) can be calculated using the formula \(m = \frac{\Delta y}{\Delta x}\) where \(\Delta y\) represents the change in the speed of sound and \(\Delta x\) represents the change in temperature. Thus, \(m = \frac{352-340}{35-15} = 0.6 m/s/°C.\)

Calculating the y-intercept (c)

Next, solve for the y-intercept, c, by substituting a given point (\(x\), \(y\)) into the linear equation, \(y = mx + c\). Using the point (15, 340), the equation becomes 340 = 0.6 * 15 + c. Solving for \(c\), we get \(c = 340 - (0.6 * 15) = 331 m/s\)

Solving for the Unknown Temperature

Finally, substitute the known speed of sound, 346m/s, into the equation, 346 = 0.6 * x + 331. Solving this equation for \(x\) (temperature), we get \(x = \frac{346 - 331}{0.6} = 25°C\)

Key concepts

Calculating Slope

Understanding the slope of a line is crucial in algebra, especially when dealing with linear relationships. The slope indicates how steep a line is and the direction it moves. Simply put, it shows how much the 'y' value, or dependent variable, will change for a one-unit change in the 'x' value, or independent variable.

To calculate the slope, also represented by the letter 'm', you use the formula:
\begin{slope_formula}\[ m = \frac{{\text{change in y}}}{{\text{change in x}}} = \frac{{\text{rise}}}{{\text{run}}} = \frac{{\text{difference in the y-coordinates}}}{{\text{difference in the x-coordinates}}} \]
\begin{slope_formula}In our scenario, we have two points representing the speed of sound at different temperatures. By using the coordinates of these points (temperature, speed), we find the difference in speed by subtracting the speeds and the difference in temperature by subtracting the temperatures of these points. Then we divide the difference in speed by the difference in temperature to get the slope of the line.

Y-intercept

The y-intercept is another fundamental concept in algebra. It is the point where the line crosses the y-axis, which means it's the value of 'y' when 'x' is zero. In a linear equation of the form \( y = mx + c \), 'c' stands for the y-intercept. It’s where your line will start on the graph if extended infinitely.

Finding the y-intercept involves using a known point on the line and the slope. After inserting these values into the linear equation and solving for 'c', you get the y-intercept. The y-intercept conveys valuable information by itself, such as the initial value of a quantity before any changes occur influenced by the slope.

Temperature and Speed of Sound Correlation

The relationship between temperature and the speed of sound is a real-world example of a linear relationship. As temperature increases, the molecules in the air move faster, which means sound waves can travel more quickly. This linear correlation can be expressed with an algebraic equation that allows us to predict the speed of sound at a given temperature, or vice versa.

Using algebra to model this correlation provides a powerful tool for prediction and understanding of how these variables interact. The slope, in this case, represents how much the speed of sound changes for each degree of temperature change. Recognizing such patterns allows us to draw broader conclusions from just two data points, as if you gather more data across different temperatures, the linear model can be fine-tuned for even more precise predictions.

Solving Linear Equations

Solving linear equations is a primary skill in algebra. These equations involve variables to the first power and can be arranged in the form \( y = mx + c \). Solving these equations is about finding the value of 'x' that makes the equation true for a given 'y'.

In the context of our example, we have already found the slope (m) and y-intercept (c) and are given a value for 'y' (the speed of sound). Inserting these into the linear equation allows us to isolate 'x' (in this case, the temperature) and find the corresponding value. The process involves rearranging the equation to solve for 'x' by subtracting the y-intercept from both sides and then dividing by the slope. This type of problem-solving typifies much of algebra, providing a method to find unknown quantities when certain information is known.