Deriving the Celsius-to-Fahrenheit Conversion Formula
Date: 12/03/2001 at 23:33:00 From: Calixta Subject: Temperature conversions What is 27 degrees above freezing in Fahrenheit? Thanks.
Date: 12/04/2001 at 01:40:05 From: Doctor Jeremiah Subject: Re: Temperature conversions Hi Calixta, It depends on what temperature scale you are converting from. Since you don't say, I am going to assume that you meant to ask what 27 degrees Celsius is when converted to Farenheit. If that's not what you meant to ask, then let me know. Let's pretend that there is a straight-line (linear) relation between the two temperature scales (and it turns out that in fact there is). If that is so, then we need two "points" to find the equation for that line. Freezing in Celsius is 0 degrees, and in Fahrenheit it is 32. Boiling in Celsius is 100 degrees, and in Fahrenheit it is 212. Farenheit Celsius -------------------- 32 0 212 100 The equation for a line that converts Celsius to Farenheit is: F = mC + b where C is the Celsius temperature m is the "slope" of the line F is the Fahrenheit temperature and b is the "intercept" of the line So if we use our two "points" to fill in this equation we get: Farenheit Celsius equation -------------------- ------------ 32 0 32 = 0m + b 212 100 212 = 100m + b Now we have two equations and two unknown values (m and b) so we can solve for these values and make an equation that converts Celsius to Farenheit: The two equations: 32 = 0m + b 212 = 100m + b The first thing to do is to make both equations equal to the same thing. The easy way to do that is to get the b terms alone on one side of the equation: 32 = b 212 - 100m = b Notice that we already know the value of b. And since both equations are equal to b, they are also equal to each other: 32 = b = 212 - 100m 32 = 212 - 100m And we can solve for m like this: 32 = 212 - 100m 32 - 212 = 212 - 100m - 212 -180 = -100m -180 / -100 = -100m / -100 (-180/-100) = (-100/-100)m 9/5 = m If we put the values for m and b into C = mF + b, we can come up with an equation that converts Celsius to Fahrenheit: F = mC + b <== m=9/5 , 32=b F = (9/5)C + 32 So this is the conversion function. We want to convert 27 degrees Celsius to Fahrenheit, so we plug 27 into this equation for C and solve for F: F = (9/5)C + 32 <== C=27 F = (9/5)27 + 32 F = 80.6 So 27 degrees Celsius is 80.6 degrees Fahrenheit. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/
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