The formula below, Newton's law of cooling, allows us to calculate the temperature of an object as it cools. T = R + (T0 − R)e-k
t T = temperature of the object after t minutes T0 = initial temperature R = temperature of the surroundings k = cooling rate (depends on the object and conditions) Suppose the temperature in Denver, Colorado was 55°F when a 5°F arctic cold front moved over the state. Using k = 0.012, how long would it take a puddle of water to freeze? (Recall that water freezes at 32°F.)
The Newton`s law of cooling: T = R + ( To - R ) * e ^(-kt ) We know that: T = 32° F ( the water freezes at 32° F ), To = 55° F, R = 5° F and the cooling rate is: k = 0.012. 33° F = 5° F + ( 55° F - 5° F ) · 2.72^(-0.012 t ) 33 - 5 = 50 · 2.72^(-0.012 t) 27 = 50 · 2.72^(-0.012 t) 2.72^(-0.012 t) = 27/50 1 / 2.72^(0.012 t) = 0.54 2.72^( 0.012 t ) = 1 : 0.54 = 1.85 0.012 t = log(base e) 1.85 = ln 1.85 = 0.6 t = 0.6 : 0.012 = 50 Answer: It would take 50 minutes.
