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Expansion of steam: When water changes to steam, its volume increases rapidly. At a normal atmospheric pressure of \(14.7\) pounds per square inch, water boils at 212 degrees Fahrenheit and expands in volume by a factor of 1700 to 1 . But when water is sprayed into hotter areas, the expansion ratio is much greater. This principle can be applied to good effect in fire fighting. The steam can occupy such a large volume that oxygen is expelled from the area and the fire may be smothered. The table below shows the approximate volume, in cubic feet, of 50 gallons of water converted to steam at the given temperatures, in degrees Fahrenheit. $$ \begin{array}{|c|c|} \hline T=\text { Temperature } & V=\text { cubic } \\ \text { feet of steam } \\ \hline 212 & 10,000 \\ \hline 400 & 12,500 \\ \hline 500 & 14,100 \\ \hline 800 & 17,500 \\ \hline 1000 & 20,000 \\ \hline \end{array} $$ a. Make a linear model of volume \(V\) as a function of \(T\). b. If one fire is 100 degrees hotter than another, what is the increase in the volume of steam produced by 50 gallons of water? c. Calculate \(V(420)\) and explain in practical terms what your answer means. d. At a certain fire, 50 gallons of water expanded to 14,200 cubic feet of steam. What was the temperature of the fire?

Step by step solution

01

Identify points for linear model

To create a linear model, we need two points from the table. Let's take the points (212, 10000) and (1000, 20000) because these encompass the entire range of data.

02

Calculate slope (m)

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting in our points, \(m = \frac{20000 - 10000}{1000 - 212} = \frac{10000}{788} \approx 12.7\).

03

Write equation of the line

Using the point-slope form of a line \(y - y_1 = m(x - x_1)\), we substitute \((x_1, y_1) = (212, 10000)\) and \(m = 12.7\). This gives us \(V - 10000 = 12.7(T - 212)\). Simplifying yields \(V = 12.7T - 1692.4 + 10000\), or \(V = 12.7T + 8307.6\).

04

Calculate the increase in volume for a 100-degree temperature rise

For part b, find how much volume increases when temperature increases by 100 degrees: \(V(T + 100) - V(T) = 12.7(T + 100) + 8307.6 - (12.7T + 8307.6) = 12.7 \times 100 = 1270\). So, the volume increases by 1270 cubic feet.

05

Evaluate V(420)

For part c, substitute \(T = 420\) into the linear model: \(V(420) = 12.7 \times 420 + 8307.6 = 14034\). This means that at 420 degrees, 50 gallons of water would convert to approximately 14,034 cubic feet of steam.

06

Find the temperature for 14,200 cubic feet of steam

For part d, set \(V = 14200\) and solve for \(T\): \(14200 = 12.7T + 8307.6\). Solving \(12.7T = 14200 - 8307.6\), gives \(12.7T = 5892.4\), and \(T \approx 464\) degrees Fahrenheit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Steam Expansion

Steam expansion is a fascinating process where water transforms into steam, significantly increasing in volume. At a standard atmospheric pressure of 14.7 pounds per square inch, water boils and turns into steam at 212 degrees Fahrenheit. This transformation results in a volume increase of about 1700 times. However, when water is introduced into environments with higher temperatures, the ratio of expansion increases greatly. This principle is crucial in certain engineering and safety applications because of its capacity to occupy large volumes.
One practical use of this phenomenally expansive property is in the context of fire fighting. By spraying water onto flames, the water turns into steam, expanding swiftly. This rapid expansion displaces oxygen from the fire's vicinity. Since fires require oxygen to sustain themselves, this can effectively smother the flames, acting as a natural extinguishing agent.

• Water to steam volume increases immensely.
• Useful for reducing fire through oxygen displacement.

Understanding this concept is vital not only for scientific curiosity but also for creating safety mechanisms in various industries.

Fire Fighting Applications

When it comes to fire fighting, understanding the properties of steam expansion can be a lifesaver. By expanding quickly, steam generated from water application can drastically reduce the fire's temperature and oxygen availability.
Firefighters utilize this principle to gain control over fires. They apply water to hot areas where the rapid change to steam helps in:

• Cooling the fire environment.
• Reducing available oxygen that fuels the fire.
• Suppressing flames through vaporization effects.

Fire fighting strategies based on steam expansion improve safety for both the personnel and surrounding environments. The ability of steam to cover expansive areas swiftly can be instrumental in stopping the spread of fires effectively. Unlocking this potential showcases the power of simple chemical transformations in everyday safety applications.

Slope Calculation

Slope calculation is an essential part of deriving linear relationships between two variables. In the context of our steam expansion exercise, it's about finding how changes in temperature affect the volume of steam.
The slope (\(m\)) helps in determining how much the volume increases per unit change in temperature. It is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \((x_1, y_1)\)and \((x_2, y_2)\)are two points on the line. For example, using points (212, 10000) and (1000, 20000) from the steam expansion data, the slope turns out to be approximately 12.7. This number reflects that for each additional degree Fahrenheit, the volume of steam increases by about 12.7 cubic feet.

• Slope represents the rate of change.
• Key for predicting volumetric expansions.

This understanding facilitates predicting future outcomes, useful for designing efficient systems and responses in practical scenarios.

Linear Equation

A linear equation is a mathematical statement that expresses a constant rate of change between two variables. In our example, the relationship between temperature (\(T\)) and steam volume (\(V\)) can be described using such an equation.
The equation derived is:\[V = 12.7T + 8307.6\]This formula allows us to compute the volume of steam for any given temperature within the studied range.

• Predict steam volume at different temperatures.
• Use to make data-driven decisions in fire fighting.

Solving this equation for specific values (e.g., finding \(V(420)\)) lets us know exactly how much volume 50 gallons of water will occupy as steam, which is crucial for planning and response in fire handling and other steam-utilizing operations. These linear insights are vital tools for engineers and safety experts alike.