$(7pts$ total$)$ Assuming $an$ air temperature $of68F$ and a living body tem$-$

perature $of98\mathrm{.}6F,$ the temperature $T(t)inFof$ a body $at$ time $t$ hours

since death $is$ given $by$:

$T(t)=68+30\mathrm{.}6e^{-kt}$

Coroners estimate time $of$ death using the rule $of$ thumb that a body cools about

$2F$ during the first hour after death and about $1F$ for each additional hour.

$(a)(1pt)$ For what value $ofk$ will the body cool $by$ approximately $2Fin$ the

first hour?

$(b)(1pt)$ Use a tangent line approximation $att=0to$ estimate the tem$-$

perature $of$ the body $24$ hours after death, using the value $ofk$ you've

found.

$(c)(1pt)$ Write a function $C(t)$ which represents the coroner's rule $of$ thumb.

$(d)(2pts)$ Compare the estimate you obtained $in$ part $(b)$ with the coroner's

rule $of$ thumb.

Which estimate $is$ more accurate, and why? How $do$ you think the coroners

obtained the rate $of1F$ per hour?

Hint: look $at$ a table $of$ values $of{T}^{\text{'}}$ for $0t24.$

$(e)(2pts)$ Suppose the thermometer being used displays values $upto2$ dec$-$

imal places, without any rounding. $Itis$ also perfectly accurate $to2$

decimal places.

$If$ the display $of$ the thermometer reads $68\mathrm{.}00F,$ what can $we$ say about the time

$of$ death?