Here's another math question about temperatures and conversions?

Question

Suppose you have designed a new thermometer called the X thermometer. On the X scale the boiling point of water is 124 X and the freezing point of water is 20 X.

At what temperature will the readings on the Fahrenheit and X thermometers be the same?

Answer 1

(124,212) and (20,32) are the points

If the relationship is linear,

x1= 124 y1= 212 x2= 20 y2= 32

solpe = (y2-y1)/(x2-x1)

= (32-(212))/(20-(124))

= (32-212)/(20-124)

slope= -180/-104

b = (y1-m*x1)

b =212-[(-180)/(-104)](124) = -2.615384615384613

Equation: y= mx +b , m = slope

Equation of the line is : y = -180/-104 x -2.6154

F=1.73077 X - 2.6154 Is the relation.

To find out when they are equal

F = 1.73077F - 2.6154

F(1-1.73077) = -2.6154

F = 2.6154 / 0.73077

F = 3.58

At this temperature , they are equal

Answer 2

Proportionality=(124-20)X/[(212-32)F]=10...

Degrees X=Degrees F * 26/45 + C, substituting for 20X and 32F and solving for C gives that C=1.5111.  then substitute that degrees X= Degrees F and solve for either. Gives that 3.5789 degrees is the same in both Fahrenheit and X thermometer scales.

Answer 3

124 X corresponds to 212 F and 20 X corresonds to 32 F.  Hence, F changes by 180° while X changes by 104°, so the slope of the line is 180/104.  The line passes through the point (20,32), so the equation is

(F - 32)/(X - 20) = 180/104 = 45/26

F - 32 = 45(X - 20)/26   <-- equation

When F = X, we have

X - 32 = 45(X - 20)/26

26X - 32(26) = 45X - 45(20)

26X - 832 = 45X - 900

68 = 19X

X = 68/19  <-- answer

As a check, when X = 68/19,

F - 32 = 45(68/19 - 20)/26

68/19 - 20 = [68 - 20(19)]/19 = (68 - 380)/19 = -312/19

F - 32 = 45(-312/19)/26 = -14040/494

F = 32 - 14040/494 = [32(494) - 14040]/494 = 1768/494 =

(26)(68)/(26)(19) = 68/19.

Answer 4

The scales are related by F-c = a(X-c), and you need to find 'a' and 'c'.

212 - c = a(124 - c)

32 - c = a(20 - c)

Subtract, to get

180 = a ( 104 )

a = 180 / 104 = 1.730769...

Now it's easy to find 'c'

c = (32 - 20a) / (1 - a) = 3.578947...

The two scales are same at 3.578947 degrees.