A radiator whose temperature is `T^(@)C` is used to heat the room in the cold weater The radiator is able maintain a room temmperature of `30^(@)C` when ouside temperature is `10^(@)C` and `15^(@)C` when outside temperature is `30^(@)C` Determine the temperature of the radiator [Assume Newton's law of cooling to be valid] .

Options

A. `85^(@)C`
B. `15^(@)C`
C. `98.6^(@)C`
D. `150^(@)C`

Answer

D

Explanation

Here the radiator is heating the room and due to conduction (through walls//windows//doors) the room gets cooled down It means in equilibrium the rate at which energy is given to room by radiator is same as the rate at which energy is lost to atmosphere From Nerwton's law of cooling
`(dQ)/(dt) = k_(1) (T -T_(0))` Where `T` is the temperature of object and `T_(0)` is the temperature of the surroundings From the law of conduction
`(dQ)/(dt) =k_(2) (T_(2) -T_(1))` Where `T_(2)` and `T_(1)` are temeratures of the bodies from which heat is transferred and to which it is transferred respectively rArr For equilibrium
`k_(1) (T -T_(0)) = k_(2) (T_(2) -T_(1))`
For first case
`k_(1) (T -30) = k_(2) [30 - (-10)] =k_(2) xx 40`
For second case
`k_(1) (T -15) =k_(2) [15 -(-30)] =k_(2) xx 45`
`rArr (T -30)/(T -15) = (40)/(45) rArr T = 150^(@)C` .