Problem 53 - Algebra 1

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. Write a linear equation that models speed of sound \(s\) in terms of air temperature \(T\).

Step-by-Step Solution

Answer

The linear equation that models speed of sound \(s\) in terms of air temperature \(T\) is \(s = 0.6T + 331\).

Step 1: Find the slope

The slope of a line is calculated by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). You can replace \(y\) with \(s\) (speed of sound) and \(x\) with \(T\) (temperature). The pairs (352,35) and (340,15) can be identified as \((y_2, x_2)\) and \((y_1, x_1)\) respectively. The slope \(m\) then becomes \(\frac{352 - 340}{35 - 15}=\frac{12}{20}=0.6\).

Step 2: Find the y-intercept

The y-intercept of the line can be found using the formula \(b = y - mx\). For either of the pairs, lets take (340,15) which refers to \((y, x)\) respectively, the y-intercept \(b\) would be \(340 - 0.6*15= 331\).

Step 3: Write the equation

The equation of speed of sound \(s\) in terms of air temperature \(T\), using the values of slope \(m\) and y-intercept \(b\) found in previous steps, then becomes \(s = 0.6T + 331\).

Key Concepts

Slope Calculation

In a linear relationship, the slope measures how much the dependent variable (in this case, the speed of sound, \(s\)) changes with a one-unit change in the independent variable (here, the air temperature, \(T\)). Calculating the slope is an essential part of establishing the equation of a line.

To compute the slope, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The symbols \(x\) and \(y\) denote our variables, with \(x\) representing temperature and \(y\) the speed of sound.

• The ordered pairs given are \((T_2, s_2) = (35, 352)\) and \((T_1, s_1) = (15, 340)\).
• Plugging these into the formula gives us \( m = \frac{352 - 340}{35 - 15} = \frac{12}{20} = 0.6 \).

Thus, the slope \(m = 0.6\) tells us that for every degree Celsius increase in temperature, the speed of sound increases by 0.6 meters per second.

Y-Intercept Determination

The y-intercept is where the line crosses the y-axis in a graph. It represents the value of \(s\) (speed of sound) when \(T\) (temperature) is zero. Finding this intercept is a crucial step in forming a linear equation.

To find the y-intercept \(b\), we use the formula \( b = y - mx \). By substituting one of our data points into this formula, we can determine \(b\). Let's choose the point \((15, 340)\).

• Using the slope \(m = 0.6\), we calculate \( b = 340 - 0.6 \times 15 = 340 - 9 = 331 \).

Therefore, the y-intercept \(b = 331\) means that at a temperature of 0°C, the speed of sound is predicted to be 331 meters per second.

Temperature and Speed Relationship

In this exercise, we are exploring the relationship between air temperature and the speed of sound. By examining how these two variables interact, we develop an equation that models this relationship.

Sound travels through air by vibrating air molecules. The speed at which sound travels can be affected by various factors, with temperature being a significant one. As temperature increases, the molecules move faster, conducive to quicker sound wave transmission.

• Using our linear equation \( s = 0.6T + 331 \), we see a clear linear relationship. As the temperature \(T\) increases, the speed \(s\) of sound in meters per second also increases.
• This equation allows us to predict speeds at varying temperatures, showcasing the direct, proportional increase of speed to temperature based on the slope \(0.6\).

Thus, understanding this relationship is vital for predicting how sound behaves in different thermal conditions, benefiting various applications like meteorology and aviation.