Question

Newton's law of cooling is , where is the temperature of an object, is in hours, is a constant ambient temperature, and is a positive constant. Suppose a building loses heat in accordance with Newton's law of cooling. Suppose that the rate constant has the value Assume that the interior temperature is , when the heating system fails. If the external temperature is , how long will it take for the interior temperature to fall to

Answer 1

Complete Question

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Answer:

The time taken  is  $t = 6.89 \ hrs$

 

Explanation:

From the question we are told that

       The value of  $k = 0.15hr^{-1}$

       The  the interior temperature is  $u(t = 0) = 70 ^oF$

         The external temperature is  $T = 11 ^oF$

          The required interior temperature is  $u_{t} = 32 ^oF$

      The newton cooling law is

                      $\frac{du}{dt} = -k (u -T)$

=>                 $\frac{du}{u-T} = - kdt$

  Now integrate both sides we have  

                 $\int\limits \frac{du}{u-T} =\int\limits - kdt$

                   $ln (u - T) = -kt + c$    

Since c is a constant lnC = c   will also give a constant so

                 $ln (u - T) = -kt + ln C$    

       =>       $\frac{(u -T)}{C} = e^{-kt}$

     =>        $u = T + Ce^{-kt}$  

          substituting value

                  $70 = 11 +Ce^{-0* 0.15}$

                  $C = 59$

Hence

        $u_t = 11 + 59 e^{-0.15 *t}$

       $32= 11 + 59 e^{-0.15 *t}$

=>     $t = -\frac{ln(\frac{21}{59} )}{0.15}$

         $t = 6.89 \ hrs$