Suppose that failure times for a device can be described by the Arrhenius$-$lognormal model:log$($t$)-\backslash$mu F$($t; Temp$)=$ P$($T $<=$ t; Temp$)=$ormwhere $\backslash$mu $=+,$x$,=11604\mathrm{.}52/($Temperature in Kelvin$),$ and $\backslash$beta $=$ E$..$

$($a$)$ Show that the logarithm of quantiles of the failure$-$time distribution is a linear function of $11604\mathrm{.}52/($Temperature in Kelvin$).$

$($b$)$Consider two temperatures Temp $>$ Tempy. Using the result in part $($a$),$ show thatthe ratio of the p quantiles at Tempy and Temp; has the relationship

t$,($Tempu$)$R$($TemP$)$R$($TemPu$)$t$,($TemP$)$

$($c$)$Use the relationship between the quantiles in part $($b$)$ to show that if R$($Temp$)=$kR$($Tempu$)$ then the life at Temp is $1/$k of the life at Tempu. For example, doublingthe rate cuts tp in half.