Three rods of identical cross-sectional area and made from the same metal form the sides of an isosceles triangle ABC right angled at B as shown in the figure. The points A and B are maintained at temperature T and `(sqrt2)T` respectively in the steady state. Assuming that only heat conduction takes place, temperature of point C will be

Options

A. `(T)/(sqrt2+1)`
B. `(T)/(sqrt2-1)`
C. `(3T)/(sqrt2+1)`
D. `(sqrt3T)/((sqrt2+1))`

Answer

C

Explanation

Let `T_(0)` be the temperature of point C and x be the length of rod AB or BC. Then, `CA= sqrt(x^(2) + x^(2))=sqrt2x`. At steady state, the rate of heat flowing from B to C = rate of heat flowing from C to A
So, `(KA (sqrt2T-T_(0)))/(x)= (KA (T_(0)-T))/(sqrt2x)`
`sqrt2(sqrt2 T-T_(0))= T_(0)-T`
By solving, `T_(0)= (3T)/((sqrt2+1))`