MHT CET202121 Sep 2021Morning ShiftMathematicsDifferential EquationsActual
A body at an unknow temperature is placed in a room which is held at a constant temperature of 30^ F . If after 10 minutes the temperature of the body is 0^ F and after 20 minutes the temperature of the body is 15^ F , then the expression for the temperature of the body at any time t is
Options
- AT =-60 e ^ -0.069 t -30
- BT =-60 e ^ -0.03010 t +30
- CT =60 e ^ -0.069 t +30
- DT =60 e ^ -0.069 t -30
Correct answer
B. T =-60 e ^ -0.03010 t +30
Step-by-step solution
We have dT dt (30- T ) aligned & dT dt =- K (30- T ) dT 30- T = - Kt & |30- T |=- kt + c aligned From given data, we write aligned & |30-0|=-10 K+c & |30-15|=-20 K+c aligned Solving (2) and (3), we get ( 30 15 )=10 ~K K = 1 10 2 Substituting value of K in eq. (2), we get 30=(-10) ( 2 10 )+ c c = 60 Thus eq. (1) becomes aligned & |30- T |= - 2 10 t + 60 & | 30- T 60 |= -0.3010 10 t =-0.03010 t & 30- T 60 = e ^ -0.03010 t T =-60 e ^ -0.03010 t +30 aligned