A body with an initial temperature `theta_(1)` is allowed to cool in a surrounding which is at a constant temperature of `theta_(0)(theta_(0)lttheta_(i))`. Assume that Newton’s law of cooling is obeyed. Let k = constant. The temperature of the body after time t is best expressed by :

Options

A. ` T = (T_(1) - T_(0))e^(-kt)` where k = `(12sigmaAT_(0)^(3))/(rrhoc)`
B. ` T = (T_(1) - T_(0))ln(kt)` where k = `(sigmaAT_(0))/(mc^(3))`
C. `T= T_(0) + (T_(1)-T_(0)) e^(-kt)` where k = `(12sigmaT_(0)^(3))/(rrhoc)`
D. `T= T_(1)e^(-kt)-T_(0)` where k = `(sigmaT_(0)^(3))/(rrhoc)`

Answer

C

Explanation

From newton's law of cooling .
`sigmaA(T^(4)-T_(0)^(4)) = ms(-(dT)/(dt))`
`implies sigma.4rhor^(2)[(T_(0)+DeltaT)^(4) - T_(0)^(4)] = rho(4)/(3)pir^(3)c(-(dT)/(dt))`
`implies ((12sigmaT_(0)^(3))/(Prc)) (T-T_(0)) = -(dT)/(dt)`
`implies K(T-T_(0))dt = -dT implies kunderset(0)overset(t)int = -underset(T.)overset(T)int(dT)/(T- T_(0))`
`implies T =T_(0) + (T_(1)- T_(0)) e^(-kt)` where k = `(12sigmaT_(0)^(3))/(rrhoc)`