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 lt theta_(1))` Assume that Newton's law of cooling is obeyed Let `k =` constant The temperature of the body after time t is best experssed by .

Options

A. `(theta_(i) -theta_(0))e^(-kt)`
B. `(theta_(i) -theta_(0)) In (kt)`
C. `theta_(0) + (theta_(i) -theta_(0))e^(-kt)`
D. `theta_(i) e^(-kt) -theta_(0)`

Answer

C

Explanation

`(d theta)/(dt) = k (theta - theta_(0))` where k = constant
`:.int_(theta_(1))^(theta)(d theta)/(theta-theta_(0))=-int_(0)^(t)k.dt`
or`[In (theta-theta_(0)]_(theta_(1))^(theta)=-kt or`
`In (theta-theta_(0)) - ln (theta_(i) - theta_(0)) = - kt`
or `(theta-theta_(0))/(theta_(i)-theta_(0))=e^(-kt)ortheta=theta_(0)+(theta_(i)-theta_(0))e^(-kt)` .