Two mercury-glass thermometer are labelled T and D respectively.On T scale, at 1 atmosphere, ice melts at -10T and pure water boils at 80T. On the D scale ice melts at 20D and pure water boils at 100D at 1 atmospheric pressure.

(a) Derive an equation relating the two temperature scales
(b) What would the temperature on D reading if on T it reads 50T?

(a) To derive an equation relating the two temperature scales, we need to find a linear relationship between the temperatures on the T and D scales.

Step 1: Find the temperature difference between the freezing and boiling points on each scale:
On the T scale, the temperature difference is 80T - (-10T) = 90T.
On the D scale, the temperature difference is 100D - 20D = 80D.

Step 2: Find the temperature difference per unit on each scale:
On the T scale, the temperature difference per unit is (90T) / (80) = 9/8 T per D.
On the D scale, the temperature difference per unit is (80D) / (90) = 8/9 D per T.

Step 3: Convert the temperature difference per unit between the scales into an equation:
Let T1 be the temperature on the T scale and D1 be the temperature on the D scale.
From Step 2, we have:
(T1 - (-10T)) / (80) = 9/8
Simplifying, we get:
T1 + 10T = 90/8
T1 = (90/8) - 10T

Similarly:
(D1 - 20D) / (90) = 8/9
Simplifying, we get:
D1 + 20D = 80/9
D1 = (80/9) - 20D

(b) To find the temperature on the D scale if it reads 50T on the T scale, we substitute T1 = 50T into the equation derived in part (a):

T1 = (90/8) - 10T
50T = (90/8) - 10T

Combining like terms:
60T = (90/8)
T = (90/8) / 60
T = 3/16

Substituting T = 3/16 into the equation for D1:

D1 = (80/9) - 20D
D1 = (80/9) - 20 * (3/16)
D1 = 80/9 - 60/9
D1 = 20/9

Therefore, if on the T scale it reads 50T, on the D scale it would read 20/9 or approximately 2.22D.