Q$2//(35-$Marks$)$

A metal rod $(k=50\frac{W}{m}.K)$ has a diameter of $1$ cm $.$ It is attached to two surfaces as shown in fig. The surfaces are $60$ cm apart. The left surface is at $200C$; the right surface is at $50C.$ The fluid surrounding the rod is at $20C$ and the heat transfer coefficient is $5\frac{W}{m^2}.K.$ If the lowest temperature in the rod is $40\mathrm{.}47C,$ then:

$($a$)$ What is the location of the lowest temperature in the rod?

$($b$)$ What is the rate of heat transfer from the rod to the fluid?