FIGURE $2\mathrm{.}47$ Battery failure data.

It is known that the average cost associated with a battery failure is $$900($including spoilage of

goods$),$ whereas a preventive replacement can be undertaken at a total cost of $$300.$ Given that

$C_f>C_p,$ the truck fleet operator wishes to consider implementing an age$-$based replacement

policy using the following model:

$C(t_p)=\frac{C_{p}R(t_p)+C_f[1-R(t_p)]}{t_{p}R(t_p)+M(t_p)[1-R(t_p)]}$

What is the optimal preventive replacement age? Consider values of $t_p=5000,10,000,15,000,$

and $20,000km.$