Meteorology Recall that water freezes at 0^∘ C(32^∘ F) and boils at 100^∘ C(212^∘ F) . (a) Find

Meteorology Recall that water freezes at 0^∘ C(32^∘ F) and...

Meteorology Recall that water freezes at $0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)$ and boils at $100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right) .$ (a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius and degrees Fahrenheit $F$ (b) Use the result of part (a) to complete the table. $\begin{array}{|c|c|c|c|c|c|}\hline C & {} & {-10^{\circ}} & {10^{\circ}} & {} & {} & {177^{\circ}} \\ \hline F & {0^{\circ}} & {} & {} & {68^{\circ}} & {90^{\circ}} & {} \\ \hline\end{array}$

Meteorology Recall that water freezes at $0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)$ and boils at $100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right) .$
(a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius and degrees Fahrenheit $F$
(b) Use the result of part (a) to complete the table.
$\begin{array}{|c|c|c|c|c|c|}\hline C & {} & {-10^{\circ}} & {10^{\circ}} & {} & {} & {177^{\circ}} \\ \hline F & {0^{\circ}} & {} & {} & {68^{\circ}} & {90^{\circ}} & {} \\ \hline\end{array}$

Key Concepts

Linear Relationships

Linear relationships model situations where changes in one variable consistently produce proportional changes in another. In the context of temperature conversion, the relationship between degrees Celsius and degrees Fahrenheit is linear, meaning it can be described by a constant rate of change over the range of temperatures.

Slope-Intercept Form

The slope-intercept form of a linear function (y = mx + b) is a crucial tool for expressing the relationship between two variables. It clearly shows how the rate of change (slope) and the starting value (y-intercept) define the entire relationship, which is used to derive a conversion equation between Celsius and Fahrenheit.

Temperature Scale Conversion

Temperature scale conversion involves creating formulas that allow one to convert a measurement on one temperature scale to another. By using known fixed points, such as the freezing and boiling points of water, a conversion equation can be derived that accurately translates temperatures from Celsius to Fahrenheit.

Interpolation Using a Linear Model

Once the conversion equation is established, interpolation allows for the calculation of intermediate temperatures. By substituting any given Celsius value into the linear model, the corresponding Fahrenheit value is obtained, enabling the completion of data tables where some values might be missing.

Transcript

00:03 Okay, we know that water freezes at 0 degrees c, and that's the same as 32 degrees f, and water boils at 100 degrees c celsius, and that's the same thing as 220 degrees f or fahrenheit.

00:19 So let's write an equation that relates c and f.

00:22 So our points here will be in the form, cf, and we have 0 is 32, and 100 is 2112.

00:37 So first we need to find the slope.

00:43 Y minus y, but f is being y here.

00:46 Because it's over in the right hand side, 212 minus 32 over 100 minus 0.

00:55 That's 180 over 100, which reduces to 9 5ths, 9 5ths equal.

01:07 And i'm going to go ahead and put it f here because the top row is f and is 32 over c minus 0.

01:18 Okay, now you can simplify this however you want.

01:20 I'm going to cross multiply like usual, and i get 9c equals 5f minus 160.

01:29 All right, so now you can either solve it for c, solve it for f, or put it in standard form where c and f are on the same side.

01:37 That's what i'm going to do.

01:38 And the reason i'm going to do that is because they're going to give us numbers to plug in that are both c and f.

01:44 All right, so we're going to fill out this chart and we're given c and f, 0 degrees and minus 10 degrees and 10 degrees, 68, 90, and 177.

02:09 Okay, so the first one, if f is zero, we get 9c minus 0 equals minus 160, z is, z is, negative 160 over 9 which is about negative 18...

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