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After heating up in a teapot, a cup of hot water is poured at a temperature of 204°F.
The cup sits to cool in a room at a temperature of 73° F. Newton's Law of Cooling
explains that the temperature of the cup of water will decrease proportionally to the
difference between the temperature of the water and the temperature of the room, as
given by the formula below:
T = T. + (To - Tale
To = the temperature surrounding the object
To = the initial temperature of the object
t = the time in minutes
T= the temperature of the object after t minutes
k = decay constant
The cup of water reaches the temperature of 180°F after 2.5 minutes. Using this
information, find the value of k, to the nearest thousandth. Use the resulting
equation to determine the Fahrenheit temperature of the cup of water, to the
nearest degree, after 4 minutes.
Enter only the final temperature into the input box.

Respuesta :

Edufirst

Answer:

  • Question 1: 0.081 min⁻¹
  • Question 2: 168ºF

Explanation:

The missing equation is:

             [tex]T=T_a+(T_0-T_a)e^{-kt}[/tex]

You are given:

  • Ta = the temperature surrounding the object = 73ºF
  • T₀ = the initial temperature of the object = 204ºF
  • t = 2.5 min
  • T after 2.5 min = 180ºF

Question 1: Find the the value of k.

Substitute the data into the equation and solve for k:

          [tex]180\º=73\º+(204\º-73\º)e^{-k\cdot 2.5min}\\ \\ 107\º=131\ºe^{-k\cdot 2.5min}\\\\-k\cdot 2.5min=\ln (107/131)\\\\k=0.081min^{-1}[/tex]

Question 2: Find the Fahrenheit temperature of the cup of water, to the nearest degree, after 4 minutes.

Subsititute into the formula and compute:

            [tex]T=73\ºF+(204\ºF-73\ºF)e^{-0.081min^{-1}\times 4min}\\ \\ T=73\ºF+(131\ºF)(0.72325024)\\\\T=168\ºF[/tex]

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