Question

Two temperature scales A and B are arbitrarily assigned the following fixed points: Ice points: 500 on A and -500 on B; Steam points: 1500 on A and 2000 on B. (a) Convert a reading of 300 on A to the corresponding degrees on B. (b) Find the temperature which is the same for both scales.

Answer 1

(a) Convert a reading of 300 on A to the corresponding degrees on B.

First, assume that between the ice and steam points there are 100 points for both scales on a certain standard reference scale. Thus, the incline of scale A will be

$$k_A=\tan\theta_A=\frac{1500-500}{100}=10,\\\space\\ k_B=\tan\theta_B=\frac{2000-(-500)}{100}=25.\\\space\\$$

The equations for temperature on scale A or B for a given value $R$ of the reference scale are

$$T_A=10R+500,\\ T_B=25R-500.$$

300^A converted to R will be

$$R=\frac{T_A-500}{10}=-20^R,$$

which corresponds to the following value on B:

$$T_B=25\cdot(-20)-500=-1000^B.$$

Thus, 300 on A is -1000 on B.

(b) Find the temperature which is the same for both scales by equating the equations:

$$T_A=T_B,\\ 10R+500=25R-500,\\ R=66.7^R.\\ T_A=T_B=1166.7.$$