# Law of cooling differential equation

• tracedinair
In summary, the conversation discusses a scenario where an object is placed in a room with a decreasing ambient temperature. The equations for Newton's Law of Cooling and the pertinent initial value problem are discussed. The conversation concludes with a suggestion to incorporate the changing ambient temperature into the equation.
tracedinair

## Homework Statement

a) An object at 200 degrees F is put in a room at 60 degrees F.The temperature of the room decreases at the constant rate of 1 degree every 10 minutes. The body cools to 120 degrees F in 30 minutes. How long will it take for the body to cool to 90 degrees F?

b) Show that the solution of the pertinent initial value problem which models the situation is:
T(t) = 60 + 140e^(kt) + [(e^(kt) - kt - 1)/(10k)]

c) Set-up an equation from which you can solve for k.

d) Set-up an equation from which the required cooling time can be found.

## Homework Equations

Newton's Law of Cooling: T'(t) = K(T(t) - T0)

Note: T is in minutes

## The Attempt at a Solution

a) This is variable seperable

dT/dt = K(T(t) - T0)

∫dT/(T(t) - T0) = ∫k dt + C

ln (T(t) - T0) = kt + C

(T(t) - T0) = ce^(kt)

T(t) = ce^(kt) + T0

At T(0) = 200, and T0 = 60

200 = ce^(K*0) + 60

c = 140

T(t) = 140e^(kt) + 60

This is where I get stuck. I'm not really sure where to go next. I'm mainly confused by the fact that room temperature is decreasing as well.

Yes, that's the problem. The differential equation you started with, dT/dt = K(T(t) - T0), assumes that the ambient temperature, T0, remains constant.

The function that represents the ambient temperature is Ta = -t/10 + 60. You need to work that into the differential equation instead of T0.

Thank you for your insight. I see where I need to go with this problem now.

## 1. What is the "Law of Cooling" differential equation?

The Law of Cooling differential equation is a mathematical model that describes the rate at which an object's temperature changes as it is exposed to a surrounding medium with a different temperature. It is commonly used to analyze heat transfer and cooling processes in various scientific and engineering fields.

## 2. What is the general form of the Law of Cooling differential equation?

The general form of the Law of Cooling differential equation is:
dT/dt = -k(T - Tm)
where dT/dt is the rate of change of temperature over time, k is a constant representing the rate of cooling, T is the temperature of the object, and Tm is the temperature of the surrounding medium.

## 3. How is the Law of Cooling differential equation derived?

The Law of Cooling differential equation is derived from Newton's Law of Cooling, which states that the rate of heat loss of an object is directly proportional to the temperature difference between the object and its surrounding medium. By using the principles of calculus and differential equations, this relationship can be represented in the form of a differential equation.

## 4. What is the significance of the constant "k" in the Law of Cooling differential equation?

The constant "k" in the Law of Cooling differential equation represents the rate of cooling, which is dependent on various factors such as the properties of the object, the medium surrounding it, and the environmental conditions. It is a crucial parameter in solving the differential equation and understanding the behavior of the cooling process.

## 5. How is the Law of Cooling differential equation applied in real-world scenarios?

The Law of Cooling differential equation has various applications in real-world scenarios, such as predicting the cooling rate of a hot cup of coffee, analyzing the cooling process of a nuclear reactor, and determining the optimal cooling time for food preservation. It is also commonly used in industries such as HVAC (heating, ventilation, and air conditioning) to design and optimize cooling systems.

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