Newton's Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and the surrounding medium. Thus, if an object is taken from an oven at 303 F and left to cool in a room at 74 F, it's temperature T after t hours will satisfy the differential equation dT/dt = k(t-74)
If the temperature fell to 206 F in 0.7 hours, what will it be in 5 hours?
To find the temperature after 5 hours using Newton's Law of Cooling, we need to solve the given differential equation:
dT/dt = k(t - 74)
To do this, we can separate the variables and integrate both sides of the equation.
∫ dT = ∫ k(t - 74) dt
Integrating both sides gives us:
T = k(t^2/2 - 74t) + C
Where C is the constant of integration.
Next, we can apply the initial condition where the temperature fell to 206 F in 0.7 hours. Let's substitute t = 0.7 and T = 206 into the equation:
206 = k(0.7^2/2 - 74 * 0.7) + C
Simplifying this equation will allow us to find the value of C:
206 = 0.245k - 51.8k + C
154.8 = -51.555k + C
Now, we can use this value of C to find the temperature T after 5 hours by substituting t = 5 into the equation:
T = k(5^2/2 - 74 * 5) + C
T = 0.5k - 370k + 154.8
Simplifying further, we get:
T = -369.5k + 154.8
To find k, we can use the initial condition once again. Substituting t = 0.7 and T = 206 into the equation:
206 = 0.245k - 51.8k + 154.8
154.8 = -51.555k + 0.245k + 206
51.7k = 51.2
k = 51.2 / 51.7
k = 0.990
Now, let's substitute k = 0.990 into our equation for T after 5 hours:
T = -369.5(0.990) + 154.8
T = -365.855 + 154.8
T ≈ 161.945
Therefore, after 5 hours, the temperature is approximately 161.945 F.