You are cooking chili. When you take it off the stove, it has a temperature of 205°F. The room temperature is 68°F and the cooling rate of the chili is r = 0.03. How long will it take to cool to a serving temperature of 95°F
To determine how long it will take for the chili to cool to a serving temperature of 95°F, we can use Newton's Law of Cooling. This law states that the rate of change of temperature of an object is proportional to the difference between the object's temperature and the surrounding temperature.
First, we need to calculate the temperature difference between the chili and the room temperature.
ΔT = Initial Temperature - Room Temperature
= 205°F - 68°F
= 137°F
Next, we need to plug the values into the formula:
dT/dt = -r * ΔT
where:
dT/dt is the rate of temperature change (cooling rate)
r is the cooling rate constant
ΔT is the temperature difference
Now, substitute the known values:
0.03 * ΔT = dT/dt
0.03 * 137°F = dT/dt
4.11°F = dT/dt
To find the time it takes to cool to the desired temperature, we integrate dT/dt with respect to t (time):
∫ dT = ∫ 4.11°F dt
T = 4.11t + C
Here, T represents the temperature, t is the time, and C is the constant of integration.
To find the value of C, we can use the initial condition:
T = 205°F, when t = 0
205 = 4.11 * 0 + C
C = 205
Now, we have:
T = 4.11t + 205
To find the time it takes for the chili to cool to 95°F, substitute T = 95°F:
95 = 4.11t + 205
Solving for t:
4.11t = 95 - 205
4.11t = -110
t = -110 / 4.11
t ≈ -26.79
Since time cannot be negative, we can conclude that it will take approximately 26.79 units of time for the chili to cool to 95°F. The specific unit of time depends on the context of the problem (e.g., minutes, hours, etc.).