newtons law of cooling

Question

A bowl of soup was brought into a room with an initial temperature of 120°. After 10 minutes, its temperature was 105°. After another 10 minutes, its temperature was 95°. Use Newton’s Law of Cooling to estimate the temperature of the room.
 
70 degrees
82 degrees
 63 degrees
 75 degrees

Answer 1

Maybe I can get you started with this.

 

Newton's law of cooling can be written in the form

 

T(t) = T(a) + (T(0)-T(a)) exp(-kt) where T(a) is the temp of the room and T(0) is the initial temp of the soup. The unknowns are T(a) and k. You have two equations

 

105 = T(a) +(120-T(a))exp(-10k)

95 = T(a) +(120-T(a))exp(-20k)

 

Now you need to solve both of these equations to find T(a) and k. You can do this by substitution of the four choices for T(a) and see which one solves both equations, or you can solve the equations. This will require you to isolate the exponentials on one side of each equation. You can use the fact that exp(-20k) is the square of exp(-10K) in order to avoid taking logarithms. In other words call B = exp(-10k), then B²= exp(-20k) and put B into both equations so that you now solve for T(a) and B

 

105 = T(a) +(120-T(a))B
95 = T(a) +(120-T(a))B²  now solve for T(a) and B -- I hope this helps