Problem 40 - Algebra 1: Concepts and Skills

Question

At sea level, the speed of sound in air is linearly related to the air temperature. If the temperature is \(35^{\circ} \mathrm{C},\) sound will travel at a rate of 352 meters per second. If the temperature is \(15^{\circ} \mathrm{C},\) sound will travel at a rate of 340 meters per second. Given the points \((35,352)\) and \((15,340),\) write in slope-intercept form the equation of the line that models this relationship.

Step-by-Step Solution

Answer

The equation that models the relationship between the speed of sound and temperature is \(y = 0.6x + 331\), where y is the speed of sound and x is the temperature.

Step 1: Calculate the Slope

Start by calculating the slope of the line (m) using the formula \(m = \frac{y2 - y1}{x2 - x1}\). Substitute the values from the two given points into the formula: \(m = \frac{352-340}{35-15}= \frac{12}{20}= 0.6\)

Step 2: Calculate the y-intercept

Next, calculate the y-intercept (b) using one point and the slope. The formula to use is \(b = y - mx\), where x and y are the coordinates of one of the points and m is the slope. Using the first point (35, 352) gives: \(b = 352 - 0.6 * 35 = 331\)

Step 3: Formulate the Final Equation

Finally, substitute the values for m and b into the equation \(y = mx + b\) to get the equation that describes the relationship between the speed of sound and temperature: \(y = 0.6x + 331\)

Key Concepts

Linear Relationship

Understanding the term 'linear relationship' is crucial when studying how two variables interact. It means that for every unit increase in one variable, there is a consistent change in another. This can be illustrated by a straight line on a graph, where one variable is on the X-axis and the other on the Y-axis.

For example, if we're considering the relationship between temperature and the speed of sound, a linear relationship suggests a uniform change in speed for every degree of temperature change. This consistency provides predictability and allows us to model the behavior of sound using a linear equation. This forms the foundation for writing equations in slope-intercept form, a popular representation used in many scientific fields.

Speed of Sound in Air

The speed of sound in air is not a constant value; it varies with temperature. Generally, warmer air provides a medium through which sound waves can travel faster. This is because the molecules in warm air are more energetic and can transmit energy from one to another more quickly compared to cooler air.

In our exercise, we assume a linear relationship between temperature and the speed of sound, meaning that there is a consistent increase in sound speed with each degree increase in temperature. Understanding how temperature affects the speed of sound is essential in many applications, from designing acoustic equipment to interpreting meteorological data.

Calculating Slope

Calculating slope is key in understanding how quickly one variable changes in relation to another in a linear relationship. The slope represents the steepness or incline of a line and is often denoted as 'm'. It's calculated as the rise over run, which translates to the change in the Y coordinates divided by the change in the X coordinates between two points on a line.

In the context of our exercise, it tells us how much the speed of sound increases for each additional degree of temperature, and this information is crucial for formulating a precise and accurate equation. If the slope is positive, it indicates a direct relationship; as temperature increases, so does the speed of sound.

Y-intercept Calculation

The y-intercept represents the point where the line crosses the Y-axis on a graph. It's a pivotal component of the slope-intercept form of an equation because it provides a starting value when the X variable is zero. The calculation involves solving for 'b' in the equation 'y = mx + b'. To find b, you use the coordinates of one point on the line and the slope you've already calculated.

For example, in the solution we derived from the speed of sound exercise, the y-intercept is the speed at which sound would theoretically travel at 0 degrees Celsius. Although this number may not always have practical significance, it is essential for creating an accurate mathematical model of the linear relationship.