Newton's Law of Cooling

See the example on page 459 of your text.
Sir Isaac Newton found that the temperature of something heated will cool down according to the function:

u(t) = Room Temp + (Heated Temp-Room Temp)ekt

where t is time
and k is some constant, depending on the material.

Example

In a 72° room, my 180° coffee will be 150° after two minutes. I like my coffee at 120°. How long should I wait?

  1. Use the info about how long it takes for my coffee to get to find k

    u(t) = Room Temp + (Heated Temp-Room Temp)ekt
    150 = 72 + (180-72)ek2
    78 = (108)e2k
    e2k = 78/108 = 13/18
    2k=ln(13/18)
    k=.5(ln(13/18)) ≈ .5(-.3254224)=-0.1627112

    So the constant of cooling for my coffee is -.1627 or so. Here is the check:

    u(t) = 72+(108)e-.1627t
     u(2)=72+108e(-.1627)(2)≈150°

  2. Solve for t

    72+108e-.1627t = 120°
    108e-.1627t = 48
    e-.1627t = 48/108=4/9
    -1627t = ln(4/9)
    t ≈ (-.8109302)/(-.1627) = 5 minutes

Challenges

  1. In a 72° room, my 180° coffee will be 150° after two minutes. How long will it take to get 75°?

  2. What is the temperture after 30 minutes?

  3. Boiling water (212° at sea level) is left in a 70° and after 5 minutes it is 180° What is the constant of cooling?

  4. Using this info from the previous question, how long will it take to have it cool to 98°?

  5. Heating is cooling in reverse. Use the same constant k as in #3. If an ice cube is placed in the same room. how long will it take to become 50°? (presume the ice is 32° when frozen).

  6. What temperature is the water after 15 minutes?
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