A rod of length l and cross-section area A has a variable thermal conductivity given by K = `alpha` T, where `alpha` is a positive constant and T is temperature in kelvin. Two ends of the rod are maintained at temperature `T_(1)` and `T_(2)` `(T_(1)gtT_(2))`. Heat current flowing through the rod will be

Options

A. `(Aalpha(T_(1)^(2)-T_(2)^(2)))/(l)`
B. `(Aalpha(T_(1)^(2)+T_(2)^(2)))/(l)`
C. `(Aalpha(T_(1)^(2)+T_(2)^(2)))/(3l)`
D. `(Aalpha(T_(1)^(2)-T_(2)^(2)))/(2l)`

Answer

A

Explanation

`-11.25 = x(tan theta)-(10x^(2))/(2xx400)(1+tan^(2)theta)`
`implies(x^(2))/(80)tan^(2)theta-x(tan theta)+((x^(2))/(80)-11.25)=0`
For real value of `'theta'`
`D ge 0`
`x^(2)-4((x^(2))/(80))((x^(2))/(80)-11.25)ge0`
`implies x^(2)le4xx625`
`implies x le 50 " "x_("max")=50m`