There's a cup of coffee made with boiling water standing at room where room temperature is 20ºC. If H(t) is the temperature of this cup of coffee at the time t, in minutes, explain what the differential equation says in everyday terms. What is the sign of k? d h d t = − k ( H − 20 )Then solve the differential equation for 90ºC in 2 minutes and how long it will take to cool to 60ºCObserving d h d t = 0 we find that H=20 this means that the function stops changing at the room temperature H=20. As t is implied to be H = 20 + A e − k t as t approaches infinity H=20.

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There&#039;s a cup of coffee made with boiling water standing at room where room temperature is 20ºC. If H(t) is the temperature of this cup of coffee at the time t, in minutes, explain what the differential equation says in everyday terms. What is the sign of k?            d      h              d      t        =  −  k  (  H  −  20  )Then solve the differential equation for 90ºC in 2 minutes and how long it will take to cool to 60ºCObserving             d      h              d      t        =  0 we find that H=20 this means that the function stops changing at the room temperature H=20. As t is implied to be   H  =  20  +  A      e          −      k      t       as t approaches infinity H=20.

Kathleen Mack · Accepted Answer

I think you can use separation of variables, so we have            d      H              d      t        =  −  k  (  H  −  20  )              d      H              H      −      20        =  −  k  d  t    ∫                    d        H                    H        −        20              =  ∫      −    k    d    t      ln  ⁡  (  H  −  20  )  =  −  k  t  +  C        e          ln      ⁡              (        H        −        20        )              =      e          −      k      t      +      C          H  −  20  =      e          −      k      t      +      C          H  −  20  =      e          −      k      t            e          C      Now, solve for       e    C  , setting t=0 gives      H    0    −  20  =      e    C  which finally gives  H  (  t  )  =  (      H    0    −  20  )      e          −      k      t        +  20      (  1  )Now we need to calculate k using equation (1) and the information given in the problem: Assume       H    0    =  100, then  90  =  (  100  −  20  )      e          −      2      k        +  20    70  =  80      e          −      2      k              70    80    =      e          −      2      k          ln  ⁡      (          70      80        )    =  −  2  k    ln  ⁡  (  7      /    8  )  =  −  2  k    k  =  −            ln      ⁡      (      8              /            7      )        2      k  ≈  0.0667656963123so  t  =            l      n      (      2      )        k      t  ≈      0.6931471805599    0.0667656963123      t  ≈  10.3817861393628So it takes about 10 minutes for the cup of coffee to cool to 60∘ C.

katdoringlo · Accepted Answer

I suppose that h is H in the equation. So, what you know is that            d      H              d      t        =  −  k  (  H  −  20  )If you remember what is a derivative, in plain words, it seems to mean that, during a short period of time, the temperature of water changes proportionally to the difference between water and room temperature.Suppose that       H    0   be the temperature at time       t    0  . For time t, you can write that            d      H              d      t        ≈            H      −              H        0                    t      −              t        0              =  −  k  (      H    0    −  20  )that is to say that, for a small interval of time,  H  ≈      H    0    −  k  (      H    0    −  20  )  (  t  −      t    0    )

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