Respuesta :
Answer:
30.06 degree Celsius at 4:00 PM.
Step-by-step explanation:
We have been given that on a hot Saturday morning while people are working inside, the air conditioner keeps the temperature inside the building at 24 degrees C. At noon the air conditioner is turned off, and the people go home. The temperature outside is a constant 35 degrees C for the rest of the afternoon. The time constant for the building is 5 hr. We are asked to find the temperature inside the building at 4:00 PM.
We will use Newton's law of cooling to solve our given problem.
[tex]T(t)=M_0+(T_0-M_0)e^{-kt}[/tex], where
[tex]T(t)[/tex] = Temperature after t hours.
[tex]M_0[/tex] = Temperature of surrounding environment,
[tex]k[/tex] = Time constant.
[tex]\frac{1}{k}=5\Rightarrow k=\frac{1}{5}[/tex]
Upon substituting our given values in Newton's cooling law, we will get:
[tex]T(t)=35+(24-35)e^{-\frac{t}{5}}[/tex]
[tex]T(t)=35-11e^{-\frac{t}{5}}[/tex]
To find the temperature at 4:00 pm, we will substitute [tex]t=4[/tex] in our formula as:
[tex]T(4)=35-11e^{-\frac{4}{5}}[/tex]
[tex]T(4)=35-\frac{11}{e^{\frac{4}{5}}}[/tex]
[tex]T(4)=35-\frac{11}{2.2255409284924674}[/tex]
[tex]T(4)=35-4.9426186052894379601[/tex]
[tex]T(4)=30.05738139471[/tex]
[tex]T(4)\approx 30.06[/tex]
Therefore, the temperature inside building would be approximately 30.06 degree Celsius at 4:00 PM.
