Question 1069666
{{{T(t)=a*e^(-kt)+90}}} makes some sense.
{{{T(t)}}} would be the body temperature in degrees at some time {{{t}}}.
 
ONE WAY TO SOLVE:
We define {{{t}}}= time in minutes after 6:00 p.m.;
substitute {{{T(0)=95.3}}} and {{{T(60)=94.9}}} to find {{{a}}} and {{{k}}},
and then use {{{T(t)=98.6}}} to find the time the victim died.
Adding the negative time we find to 6:00 p.m. gives us the murder time.
It gets easier if we solve {{{T(t)=a*e^(-kt)+90}}} for what we want to start,
and then just plug stuff in.
{{{a=(T-90)/e^(-kt)}}}
{{{k=-(1/t)ln((T-90)/a)}}}
{{{t=-(1/k)ln((T-90)/a)}}}
At 6:00 p.m. {{{t=0}}}<-->{{{-kt=0}}} and {{{T=95.3}}} , so
{{{a=(95.3-90)/e^0}}}-->{{{a=5.3/1}}}-->{{{highlight(a=5.3)}}}
At 7:00 p.m. {{{t=60}}} and {{{T=94.9}}} , so
{{{k=-(1/60)ln((94.9-90)/5.3)}}}-->{{{k=-(1/60k)ln(4.9/5.3)}}}-->{{{k=-(1/60)(-0.7850)}}}-->{{{highlight(k=0.001308)}}}
At the time the victim died, {{{T=98.6}}} , so
{{{t=-(1/60)ln((98.6-90)/5.3)}}}-->{{{t=-(1/60)ln(8.6/5.3)}}}-->{{{t=-(1/60)0.4840}}}-->{{{highlight(t=-370)}}} (rounded to the nearest minute)
Since {{{370minutes=6hours+11minutes}}} ,
time of death = {{{"6:00 PM"-"6:11"=highlight("11:49 AM")}}} .
 
EXPLANATION AND ALTERNATE WAY TO SOLVE:
You may have heard in class that
{{{y=e^(-kx)}}} for some {{{k>0}}} , {{{graph(200,200,-0.1,0.9,-0.1,0.9,0.8*e^(-(sqrt(5x)^2)))}}}
models exponential decay;
that it asymptotically approaches {{{y=0}}},
or even that {{{lim(x->infinity,e^(-kx))=0}}} .
 
{{{T(t)=a*e^(-kt)+90}}} (with some {{{a>0}}} and {{{k>0}}} too) makes sense.
It would have {{{T(0)=a*e^0+90=a*1+90=a+90}}} ,
and would decrease towards {{{90}}} , which is the ambient temperature in degrees.
Nothing else is specified  about that function,
so we will say that
{{{t}}}= time since 6:00 p.m., in minutes.
So, {{{T(0)=95.3}}} tells us that
{{{a+90=95.3}}} --> {{{a=95.3-90}}} ---> {{{highlight(a=5.3)}}}
An hour later, {{{t=60}}} ,
and {{{T(60)=94.9}}} tells us that
{{{a*e^(-60k)+90=94.9}}} --> {{{5.3*e^(-60k)=94.9-90}}} --> {{{5.3*e^(-60k)=4.9}}}
--> {{{e^(-60k)=4.9/5.3}}} --> {{{-60k=ln(0.9245)}}} --> {{{-60k=-0.07850}}}
--> {{{k=(-0.07850)/(-60)}}} --> {{{highlight(k=0.001308)}}}
Now we have the function
{{{T(t)=5.3e^(-0.001308t)+90}}}
and we want to find {{{t}}} for {{{T(t)=98.6}}}
{{{98.6=5.3e^(-0.001308t)+90}}}
{{{98.6-90=5.3e^(-0.001308t)}}}
{{{8.6=5.3e^(-0.001308t)}}}
{{{8.6/5.3=e^(-0.001308t)}}}
{{{ln(8.6/5.3)=-0.001308t}}}
{{{t=ln(8.6/5.3)/(-0.001308)}}}
{{{highlight(t=-370)}}} (rounded to the nearest minute)
Since {{{370minutes=6hours+11minutes}}} ,
time of death = {{{"6:00 PM"-"6:11"=highlight("11:49 AM")}}} .