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Derive the Function of Celsius to Fahrenheit

Date: 09/22/97 at 21:34:54
From: Ry Larrandson
Subject: Derive the function of Celsius to Fahrenheit

I want to know how you get the equation of Celsius to Fahrenheit 
and vice versa. How would you graph that function? Also, how do you 
use that graph to convert F degrees to C degrees and C degrees to 
F degrees?

If you can help, please do. :-)

Date: 09/23/97 at 03:48:46
From: Doctor Pete
Subject: Re: Derive the function of Celsius to Fahrenheit


In order to know how to change between units of Celsius and 
Fahrenheit, you need to know two points on each scale that are the 
same temperature. For example, Americans know standard body 
temperature to be 98.6 F.  But what is this value in C?  

Well, we don't know the corresponding value in Celsius so well, but 
there are two temperatures we are fairly familiar with in both scales: 
the freezing and boiling points of water (at standard atmospheric 
pressure). The Fahrenheit values are familiar to Americans:  32 F and 
212 F, respectively.  And we know Celsius, because that's easy:  0 C 
and 100 C (in fact, this is where the Celsius scale is derived from).  
So we have

     0 C = 32 F
   100 C = 212 F.

To get from these two equations to our conversion equation, we make 
the observation that both Celsius and Fahrenheit are *linear*. That 
is, if we take the temperature halfway between 0 and 100 C (50 C), it 
is also the same temperature if we take the point halfway between 32 
and 212 F (122 F). This is also true with 1/3rd, 2/5ths, etc. etc.  

Now say we want to convert from C to F. If we have x degrees Celsius, 
the corresponding value in F is y = A*x + B, where A and B are 
unknowns. (Note that if we replace A and B with m and b, we get the 
familiar slope-intercept form of a line - hence the "linearity".)  
To determine A and B, we use the reference points we mentioned 
earlier, so:

     32 F = A*(0 C) + B = B,

which implies B = 32 F.  Then

    212 F = A*(100 C) + 32 F,

so A = 1.8 F/C.  Therefore, x degrees Celsius is equivalent to 
1.8 x + 32 degrees Fahrenheit. To go the other direction, we simply 
invert the equation, so y degrees Fahrenheit is equivalent to 
(y - 32)/1.8 degrees Celsius. So 98.6 F = 37 C, exactly.  

And the recent heat wave in Los Angeles has about the flavor of 32 C.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Functions
Middle School Temperature

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