Question 1110574
The initial temperature difference in degrees Fahrenheit is
{{{65-19=46}}} .
Nature wants to get to equilibrium,
and it heads there at a rate
proportional to the magnitude of the deviation from equilibrium.
Without talking about differential equations,
the difference in temperature will undergo "exponential decay."
The temperature difference will be an exponential function of time.
Because it has been found so useful for calculus,
we like to write exponential functions using the irrational number {{{e}}} .
With
{{{t}}}=time in minutes since the thermometer was brought into a room,
and {{{F}}} being the thermometer reading in degrees Fahrenheit,
the temperature difference function will be
{{{65-F=46*e^(-kt)}}} , with a positive constant {{{k}}} to be determined.
If we want, we can solve for {{{F}}} to get
{{{F=65-46*e^(-kt)}}} .
 
To determine {{{k}}} we substitute {{{t=1}}} and {{{F=30}}} into
{{{65-F=46*e^(-kt)}}} to get
{{{65-30=46*e^(-k)}}} ,
{{{35=46*e^(-k)}}}
{{{35/46=e^(-k)}}}
{{{ln(35/46)=-k}}}
{{{k=-ln(35/46)=about0.27329}}} (rounding to 5 decimal places).
 
So, the thermometer reading is predicted by
{{{highlight(F=65-46*e^(-0.27329t))}}} .
 
Four (4) minutes after the thermometer is brought into the room, {{{t=4}}} , and
{{{F=65-46*e^(-0.27329*4)=65-46*e^(-1.09316)=65-46*0.33516=65-15.4=49.6}}} .
The temperature reading 4 minutes after the thermometer is brought into the room
is expected to be {{{highlight(50)}}} degrees Fahrenheit.