SOLUTION: A thermometer reading 19 Fahrenheit is brought into a room whose temperature is 65 F. After a minute, the thermometer reading is 30 F. � Set up an equation for predicting the therm

Question 1110574: A thermometer reading 19 Fahrenheit is brought into a room whose temperature is 65 F. After a minute, the thermometer reading is 30 F. � Set up an equation for predicting the thermometer reading at any time t in minutes. � Find the temperature 4 minutes after the thermometer is brought into the room.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The initial temperature difference in degrees Fahrenheit is
.
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 .
With
=time in minutes since the thermometer was brought into a room,
and being the thermometer reading in degrees Fahrenheit,
the temperature difference function will be
, with a positive constant to be determined.
If we want, we can solve for to get
.

To determine we substitute and into
to get
,

(rounding to 5 decimal places).

So, the thermometer reading is predicted by
.

Four (4) minutes after the thermometer is brought into the room, , and
.
The temperature reading 4 minutes after the thermometer is brought into the room
is expected to be degrees Fahrenheit.
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