Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

The amount of heat \(H\) (in joules) required to convert one gram of water into vapor is linearly related to the temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) of the atmosphere. At \(10^{\circ} \mathrm{C}\) this conversion requires 2480 joules, and each increase in temperature of \(15^{\circ} \mathrm{C}\) lowers the amount of heat needed by 40 joules. Express \(H\) in terms of \(T\).

Step-by-Step Solution

Answer

The equation is \( H = -\frac{8}{3}T + 2506.67 \).

Step 1: Understanding the Linear Relationship

We are told that the heat required is linearly related to temperature. This means we can express the relationship using the equation \( H = mT + b \) where \( m \) is the slope of the line, \( T \) is the temperature, and \( b \) is the y-intercept.

Step 2: Identify Known Values

We know that at \( T = 10^{\circ} \mathrm{C} \), \( H = 2480 \) joules. This gives us the point (10, 2480). Also, the problem states that each increase in temperature of \( 15^{\circ} \mathrm{C} \) decreases \( H \) by 40 joules.

Step 3: Calculate the Slope

The slope \( m \) is calculated as the change in \( H \) divided by the change in \( T \). Since an increase of \( 15 \) degrees results in a decrease of 40 joules, the slope \( m = \frac{-40}{15} = -\frac{8}{3} \).

Step 4: Use a Known Point to Find y-intercept

We use the point \((10, 2480)\) to find \( b \). Substitute \( H = 2480 \), \( T = 10 \), and \( m = -\frac{8}{3} \) into the equation \( 2480 = -\frac{8}{3}(10) + b \). Solving for \( b \), we get \( b = 2480 + \frac{80}{3} = 2480 + 26.67 = 2506.67 \).

Step 5: Write the Final Equation

Now we have all the components to write the equation. The linear equation for the heat in terms of temperature is \( H = -\frac{8}{3}T + 2506.67 \).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental in grasping the linear relationship between two variables. This form of a linear equation is represented as \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( m \) represents the slope of the line.
• \( x \) is the independent variable.
• \( b \) is the y-intercept, or the value of \( y \) when \( x \) equals zero.

In the given exercise, by expressing the heat required \( H \) in terms of temperature \( T \), we can easily use the slope-intercept form, \( H = mT + b \).
Utilizing this form allows us to predict how much heat is needed as the atmospheric temperature changes.
This is mainly because the slope, \( m \), tells us how \( H \) changes with each degree of temperature change.

Temperature and Heat Relationship

The relationship between temperature and heat in this exercise is a classic example of a linear relationship. As temperature increases, the heat required for conversion decreases.
This is because temperature and heat are inversely related here.
In practical terms, this means that:

• Higher atmospheric temperatures require less heat to convert water into vapor.
• For each increment in temperature, there is a consistent decrease in the amount of joules needed.

This relationship is visually easy to identify in a graph where the slope points downwards, illustrating the decrease in heat with an increase in temperature. This visual insight can help understand the practical implications of climate and thermodynamic principles.

Calculation of Slope

Calculating the slope is essential in understanding the depth of a linear relationship. The slope, \( m \), is determined by dividing the change in the dependent variable by the change in the independent variable.
For the exercise, we determine the slope \( m \) as follows:

• We know an increase of \( 15^{\circ} \) in temperature reduces heat by 40 joules.
• Thus, the slope \( m = \frac{-40}{15} = -\frac{8}{3} \).

This negative slope indicates that as the temperature goes up, the required heat goes down.
The calculation of the slope is pivotal because it shapes how our equation can predict heat requirements at various temperatures.

Solving Linear Equations

Solving linear equations is like piecing together a puzzle where each component of the equation holds vital information.
In our exercise:

• First, the slope \( m = -\frac{8}{3} \).
• Then use the known point (10, 2480) to solve for the y-intercept \( b \).

By substituting into the equation \( 2480 = -\frac{8}{3}(10) + b \), you find \( b = 2506.67 \).
Now, you can write the complete linear equation: \( H = -\frac{8}{3}T + 2506.67 \).
Understanding this process allows us to formulate equations that provide accurate estimates of one variable based on changes to another, serving as a powerful tool in mathematics and science.