The model describing the temperature in a rod where the temperature at the left end is zero and where there is heat transfer from the right boundary into the external medium is
A) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h 0 , u ( x , 0 ) = f ( x )$
B) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = h u ( L , t ) , h 0 , u ( x , 0 ) = f ( x )$
C) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h 0 , u ( x , 0 ) = f ( x )$
D) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = h u ( L , t ) , h 0 , u ( x , 0 ) = f ( x )$
E) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h 0 , u ( x , 0 ) = f ( x )$

Question

The model describing the temperature in a rod where the temperature at the left end is zero and where there is heat transfer from the right boundary into the external medium is
A) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$
B) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$
C) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$
D) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$
E) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$

Quizplus · Accepted Answer

The correct model for describing the temperature in a rod with the given conditions is the heat equation, which is represented by $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t }$. The boundary condition at the left end is $u(0, t) = 0$, indicating the temperature is zero. The heat transfer from the right boundary into the external medium is modeled by $\frac { \partial u } { \partial x } (L, t) = - h u (L, t)$, where $h > 0$ represents the heat transfer coefficient. This setup is accurately described in option C.

The Model Describing the Temperature in a Rod Where the Temperature