Answer:
(a) for 43.75 mm rod, the temperature is 121.97 ⁰C
(b) for 87.50 mm rod, the temperature is 80.17 ⁰C
(c) for 175.00 mm rod, the temperature is 53.46 ⁰C
Explanation:
Given;
L = 175 mm = 0.175 m
D = 5mm = 0.005
= 200°C
T∞ = 20°C
Heat transfer coefficient h = 30 W/m²·K
Thermal conductivity of brass K = 133 W/m.°C

where;
T is the temperature of the rod at different casting distance,
![T_X =\frac{Cosh[m(L-X)]+(h/mk)Sinh[m(L-X)]}{Cosh(mL) +(h/mk)Sinh(mL)}](https://tex.z-dn.net/?f=T_X%20%3D%5Cfrac%7BCosh%5Bm%28L-X%29%5D%2B%28h%2Fmk%29Sinh%5Bm%28L-X%29%5D%7D%7BCosh%28mL%29%20%2B%28h%2Fmk%29Sinh%28mL%29%7D)

Part (a) for 43.75 mm rod
![T_{43.75} =\frac{Cosh[13.43(0.175-0.04375)] +(0.0168)Sinh[13.43(0.175-0.04375)]}{Cosh(13.43*0.175)+(0.0168)Sinh(13.43*0.175)} \\\\T_{43.75} = \frac{2.9999+0.0475}{5.2918+0.0875} = 0.5665](https://tex.z-dn.net/?f=T_%7B43.75%7D%20%3D%5Cfrac%7BCosh%5B13.43%280.175-0.04375%29%5D%20%2B%280.0168%29Sinh%5B13.43%280.175-0.04375%29%5D%7D%7BCosh%2813.43%2A0.175%29%2B%280.0168%29Sinh%2813.43%2A0.175%29%7D%20%5C%5C%5C%5CT_%7B43.75%7D%20%3D%20%5Cfrac%7B2.9999%2B0.0475%7D%7B5.2918%2B0.0875%7D%20%3D%200.5665)

Part (b) for 87.50 mm rod
![T_{87.5} =\frac{Cosh[13.43(0.175-0.0875)] +(0.0168)Sinh[13.43(0.175-0.0875)]}{Cosh(13.43*0.175)+(0.0168)Sinh(13.43*0.175)} \\\\T_{87.5} = \frac{1.7735+0.0246}{5.2918+0.0875} = 0.3343](https://tex.z-dn.net/?f=T_%7B87.5%7D%20%3D%5Cfrac%7BCosh%5B13.43%280.175-0.0875%29%5D%20%2B%280.0168%29Sinh%5B13.43%280.175-0.0875%29%5D%7D%7BCosh%2813.43%2A0.175%29%2B%280.0168%29Sinh%2813.43%2A0.175%29%7D%20%5C%5C%5C%5CT_%7B87.5%7D%20%3D%20%5Cfrac%7B1.7735%2B0.0246%7D%7B5.2918%2B0.0875%7D%20%3D%200.3343)

Part (c) for 175.00 mm rod
![T_{175} =\frac{Cosh[13.43(0.175-0.175)] +(0.0168)Sinh[13.43(0.175-0.175)]}{Cosh(13.43*0.175)+(0.0168)Sinh(13.43*0.175)} \\\\T_{175} = \frac{1+0}{5.2918+0.0875} = 0.1859](https://tex.z-dn.net/?f=T_%7B175%7D%20%3D%5Cfrac%7BCosh%5B13.43%280.175-0.175%29%5D%20%2B%280.0168%29Sinh%5B13.43%280.175-0.175%29%5D%7D%7BCosh%2813.43%2A0.175%29%2B%280.0168%29Sinh%2813.43%2A0.175%29%7D%20%5C%5C%5C%5CT_%7B175%7D%20%3D%20%5Cfrac%7B1%2B0%7D%7B5.2918%2B0.0875%7D%20%3D%200.1859)
