Newtons Law of Cooling - Algebraic Equation

[Ruby123]

Hi, I am a newbie here and sooo happy I might have found help. I am stumped!!
I am working on Newtons Law of Cooling - I have a snipped example below but I have gotten confused at the point circled in red and no idea how they got the answer where the arrow is pointing. I have put my actual algebraic equation below, which I am trying to calculate. If anyone could provide an explanation (feel free to dumb it down) - a big thank you

• t = time taken for the cooling
• T(t) = the temperature of the object at time t
• Ts = the surrounding (ambient temperature)
• T0 = the initial temperature of the beverage
• K =decay constant

• T(t) = Ts + (T0 – Ts)e^(-kt)
• T(t) = 22 + (71 – 39.6) e^(-kt}
• To find K when t =5, T(t) = 39.6
• T(t) = 39.6 = 22+31.4 (e)^(-5t)
• 17.6 = 31.4 (e)^(-kt)
• 17.6/31.4 = (e)^(-5k)
• ln (17.6/31.4) = ln (e)^(– 5k)
• ln (17.6/31.4) = -5k
• k ≈

 

[Deleted member 4993]

Hi, I am a newbie here and sooo happy I might have found help. I am stumped!!
I am working on Newtons Law of Cooling - I have a snipped example below but I have gotten confused at the point circled in red and no idea how they got the answer where the arrow is pointing. I have put my actual algebraic equation below, which I am trying to calculate. If anyone could provide an explanation (feel free to dumb it down) - a big thank you

• t = time taken for the cooling
• T(t) = the temperature of the object at time t
• Ts = the surrounding (ambient temperature)
• T0 = the initial temperature of the beverage
• K =decay constant

View attachment 27692

• T(t) = Ts + (T0 – Ts)e^(-kt)
• T(t) = 22 + (71 – 39.6) e^(-kt}
• To find K when t =5, T(t) = 39.6
• T(t) = 39.6 = 22+31.4 (e)^(-5t)
• 17.6 = 31.4 (e)^(-kt)
• 17.6/31.4 = (e)^(-5k)
• ln (17.6/31.4) = ln (e)^(– 5k)
• ln (17.6/31.4) = -5k
• k ≈

Are you implying that you do not know "how to calculate ln(17.6/31.4)"?

 

[Ruby123]

Are you implying that you do not know "how to calculate ln(17.6/31.4)"?

I can calculate the equation ( I have attached all my calculations now) but my understanding of how the ln and e cancel each other out is sketchy and my final calculation of my equation is not working so I am assuming I am doing something wrong.

 

[Deleted member 4993]

I can calculate the equation ( I have attached all my calculations now) but my understanding of how the ln and e cancel each other out is sketchy and my final calculation of my equation is not working so I am assuming I am doing something wrong.
View attachment 27693

The last line of the left-box is INCORRECT. You write:

T(t) = 22 + 31.4e^{-0.1157817987} ....... this is incorrect. You have forgotten to include t.

It should be:

T(t) = 22 + 31.4e^{(-0.1157817987* t)}

correct that and correct the following calculations.

 

[Ruby123]

The last line of the left-box is INCORRECT. You write:

T(t) = 22 + 31.4e^{-0.1157817987} ....... this is incorrect. You have forgotten to include t.

It should be:

T(t) = 22 + 31.4e^{(-0.1157817987* t)}

correct that and correct the following calculations.

Thank you, I spent so much time trying to work out what I was doing wrong, I don't know how I dropped that - makes sense now