Pre Calculus Question

Question

Coffee is poured at 145ºF into a travel cup. After 25 minutes in a 70º room the coffee has cooled to 137ºF. I accidentally left the coffee on the counter when I left. What is the temperature after 4 hours? after 5 hours? how long before the temperature is below 90º?

Show the process.

Answer 1

Given:

T(0) = 145°F

T_a = 70°F

T(25) = 137°F

Find:

T(240) = ?°F

T(300) = ?°F

Assume:

The coffee system will obey the Newton Law's of Cooling equation:

dT/dt = -k ( T - T_a )

Solution:

The above equation is a separable, first order differential equation that can be solved like so:

dT / ( T - T_a ) = -k dt

Integrating both sides will get to the final solution needed. The bounds of integration are simple in nature but interpreting them will require some explanation.

The time bounds will be from 0 to t, simple enough.

The temperature bounds will be similar but not the exact same, because it wouldn't make sense for temperature to be bounded from 0° to t°; alternatively, it will be bounded from the temperature at time 0, T(0), to the temperature at time t, T(t).

∫_T(0)^T(t) dT / ( T - T_a ) = -k ∫₀^t dt

ln | ( T(t) - T_a ) / ( T(0) - T_a ) | = -kt As a side derivation, k = 1 / t * ln | ( T(0) - T_a ) / ( T(t) - T_a ) |

( T(t) - T_a ) / ( T(0) - T_a ) = e^-kt

T(t) - T_a = ( T(0) - T_a ) * e^-kt

T(t) = T_a + ( T(0) - T_a ) * e^-kt

Utilizing the derivation for the constant k from above

k = 1 / 25 * ln | ( T(0) - T_a ) / ( T(25) - T_a ) | = 1 / 25 * ln | ( 145 - 70 ) / ( 137 - 70 ) |

k ≈ 0.00451

And finally,

T(240) = 70 + ( 145 - 70 ) * e^{-(0.00451)(240)}

∴ T(240) ≈ 95.40°F

T(300) = 70 + ( 145 - 70 ) * e^{-(0.00451)(300)}

∴ T(300) ≈ 89.37°F

If you've been wondering why I used 240 and 300 for 4 hours and 5 hours, I converted the hours to minutes since the first unit that was used was 25 minutes so every time unit afterwards, I assumed and derived with minutes as the time unit. This whole problem can be done with the time unit being hours, every time mark that I used in this problem will be scaled down by a factor of 60 (because of 60 minutes for every 1 hour).

I hope this helps! Please message me in the comments with any questions, comments, or concerns!