## Real World Applications – Algebra I

### Topic

How Boiling Point is Different at Different Altitudes

### Student Exploration

At areas that are at or close to sea level, the boiling point of water is at 212 degrees Fahrenheit. The higher above sea level you’re at, the lower the boiling point is. Why do you think that is?

Since the air is thinner at higher altitudes, the boiling point of water is lower. Scientists have used an approximate formula to find the boiling point of water at high altitudes. The formula is:

\begin{align*}D^2 +520D=H\end{align*}

“\begin{align*}D\end{align*}” represents the degrees below 212 degrees in which water will boil, and “\begin{align*}H\end{align*}” represents the altitude above sea level.

The city of Custer, in South Dakota, sits at 5,300 feet above sea level. We can use this formula to find the point at which water will boil.

\begin{align*}D^2 +520D &= H\\ D^2+520D &=5300\\ D^2 +520D-5300 &=0\end{align*}

We’ve set this equation equal to zero to find the “\begin{align*}D\end{align*}” intercepts of this quadratic equation, or to solve for “\begin{align*}D\end{align*}.” Now, we will factor this quadratic. We know that the factors of 5300 the have a difference of 520 are 530 and 10, so we will use these numbers in our factors.

\begin{align*}D^2+520D-5300 &=0\\ (D+530)(D-10) &=0\end{align*}

Now that we’ve factored our quadratic expression, we can set each factor equal to zero and find the “\begin{align*}D-\end{align*}intercepts.”

\begin{align*}D + 530 = 0 \ and \ D - 10 &= 0\\ D = - 530 \ and \ D &= 10\end{align*}

What do you think these two answers mean? Do both of these answers make sense to what we were trying to find?

For finding the boiling points at high altitudes, the answer, - 530 doesn’t make sense because \begin{align*}D\end{align*} represents the degrees below 212 that water will boil. \begin{align*}212-(-530)=842\end{align*}, which doesn’t make sense! On the other hand, \begin{align*}D = 10\end{align*} makes sense because this means that at Custer, water will boil 10 degrees below 212, or 212 – 10 or 202 degrees.

One scientist created a shortcut and said that the boiling point will be 1 degree lower for every 500ft increase. Does this work with Custer?

Since Custer is at 5,300 feet, we divide this by 500 feet. 5300 divided by 500 is 10.6, meaning that, according to this rule, water will boil 10.6 degrees lower than 212. This is pretty close! But would this rule hold true for areas at different altitudes?

### Extension Investigation

Try looking up different areas around the world and find if the shortcut holds true for the city you choose. Use both the formula to find the boiling point of water, and then use the shortcut. Try looking at http://en.wikipedia.org/wiki/List_of_highest_towns_by_country to find some of the world’s highest towns by country.

### Resources Cited

http://en.wikipedia.org/wiki/List_of_highest_towns_by_country

### Connections to other CK-12 Subject Areas

#### Chemistry

- Boiling Points
- Pressure versus Temperature