## Real World Applications – Algebra I

### Topic

How do we convert the temperature?

### Student Exploration

In different countries all around the world, people define the temperature in two different ways. There are two different sets of scales that people use: degrees in Celsius and Fahrenheit. In the United States, all of the temperature readings are in Fahrenheit. What if you were to go to a different country? How can you convert the temperature to units that you can understand?

Nowadays, there are apps and internet calculators for people to figure this out, but they all use a specific formula to figure this out. The formula to convert Celsius to Fahrenheit is \begin{align*}F = \left ( \frac{9}{5} \right )C + 32\end{align*}.

On one summer day in Puerto Rico, the temperature read 34 degrees Celsius. If you’re visiting from the United States and not used to the Celsius reading, what does this mean to you? Let’s use the formula to convert to Fahrenheit.

\begin{align*}F & = \left ( \frac{9}{5} \right )C + 32\\ F & = \left ( \frac{9}{5} \right )(34) + 32\\ F & = \left ( \frac{306}{5} \right ) + 32\\ F & = 61.2 + 32\\ F & = 93.2\end{align*}

So, if the temperature in Puerto Rico is 34 degrees Celsius, it’s 93.2 degrees Fahrenheit! That’s pretty hot!

In one of the concepts, you also learned how to translate this algebraic equation into words. The above equation reads, to find the temperature in Fahrenheit, you take the sum of 32 and the \begin{align*}\frac{9}{5}\end{align*} times the temperature in degrees Celsius.

There is also a formula you can use to convert from Fahrenheit to Celsius: \begin{align*}C = \left ( \frac{5}{9} \right )(F - 32)\end{align*}. The translation of this equation into words is: To find the temperature in degrees Celsius, you take the product of 5/9 and the difference between the degrees in Fahrenheit and 32.

If the weather in New York on a cold winter morning is 20 degrees Fahrenheit, how can you explain the temperature to someone that isn’t aware of the temperature in Fahrenheit? Let’s use the formula.

\begin{align*}C & = \left ( \frac{5}{9} \right )(F - 32)\\ C & = \left ( \frac{5}{9} \right )(20 - 32)\\ C & = \left ( \frac{5}{9} \right )(- 12)\\ C & = - \frac{60}{9}\\ C & = - 6.7 \ degrees!\end{align*}

Let’s look at the last equation that converts degrees in Fahrenheit to degrees in Celsius. We can find the intercepts of this linear equation two different ways that are taught in this flexbook: Finding the intercepts by substitution and finding the intercepts using the cover-up method. Which method would work best for the formula, \begin{align*}C = \left ( \frac{5}{9} \right )(F - 32)\end{align*}?

The best method would be to use substitution. (We wouldn’t want to use the cover-up method unless the equation is in standard form). If we substituted 0 into the equation for \begin{align*}F\end{align*}, we can find the \begin{align*}C\end{align*} value. This would mean that we can find the equivalent temperature in Celsius to 0 degrees Fahrenheit.

\begin{align*}C & = \left ( \frac{5}{9} \right )(F - 32)\\ C & = \left ( \frac{5}{9} \right )(0 - 32)\\ C & = \left ( \frac{5}{9} \right )(- 32)\\ C & = - 17.8 \ degrees\end{align*}

Try doing the same to find the equivalent to 0 degrees Celsius. What did you get? Which equation would you use?

You can actually use either equation. You should get the same answer. Since we’ve been working with the second formula, let’s use that one to find the equivalent to 0 degrees Celsius.

\begin{align*}C = \left ( \frac{5}{9} \right )(F - 32)\end{align*}

\begin{align*}0 = \left ( \frac{5}{9} \right )(F - 32)\end{align*} First, multiply both sides by \begin{align*}\frac{9}{5}\end{align*}, which is the reciprocal to cancel out the fraction.

\begin{align*}0 = F - 32\end{align*} Now add 32 to both sides

\begin{align*}32 = F\end{align*}

This means that 32 degrees Fahrenheit means the same thing as 0 degrees Celsius.

How would we get either formula in standard form and then use the cover-up method to find the intercepts? What would these intercepts mean?

Let’s use the first formula this time, just to change things up a bit. \begin{align*}F = \left ( \frac{5}{9} \right )C + 32\end{align*}

First, we’ll multiply every term by 5, since we can get rid of the fraction.

\begin{align*}5F = 9C + 160\end{align*} Now we’ll subtract \begin{align*}9C\end{align*} from both sides.

\begin{align*}5F – 9C = 160\end{align*}

Now we can use the cover-up method to find either intercept. We’ll find the C intercept together. Cover up “\begin{align*}5F\end{align*}”

\begin{align*}- 9C = 160 C = -17.8 \ degrees\end{align*}

Now we can do the same and cover up the “\begin{align*}-9C\end{align*}” to find the “\begin{align*}F\end{align*}” intercept.

\begin{align*}5F = 160 F = 32 \ degrees\end{align*}