Newton's Law of Cooling A container of hot liquid is placed...

Newton's Law of Cooling A container of hot liquid is placed in a freezer that is kept at a constant temperature of $20^{\circ} \mathrm{F}$ . The initial temperature of the liquid is $160^{\circ} \mathrm{F}$ . After 5 minutes, the liquid's temperature is $60^{\circ} \mathrm{F}$ . How much longer will it take for its temperature to decrease to $30^{\circ} \mathrm{F} ?$

Key Concepts

Differential Equations

Newton's Law of Cooling is expressed as a first-order linear differential equation, which describes the rate of change of temperature over time. Solving this differential equation leads to an exponential function that incorporates both the initial conditions and the ambient temperature.

Initial Conditions

To find the specific solution to the cooling process, initial conditions such as the starting temperature of the object and the ambient temperature must be incorporated. These conditions are essential for determining the constant of proportionality in the exponential decay model, which directly influences the cooling rate.

Exponential Decay

The solution to Newton's Law of Cooling is an exponential function, where the temperature difference between the object and its environment decreases exponentially over time. This concept explains how quickly the object cools down depending on factors like the cooling constant.

Newton's Law of Cooling

This law describes how the temperature of an object changes over time as it approaches the temperature of its surrounding environment. It models the cooling (or warming) process by stating that the rate of change in the object's temperature is proportional to the difference between its own temperature and the ambient temperature.

Transcript

0:00 Hello.

00:04 As we read the text of the problem, we see that we need to apply newton's law of cooling that says that the rate of the change of the temperature is going to be proportional to the difference of the temperature and the constant temperature at which is being kept and in the text this is the 20.

00:31 Degree fahrenheit.

00:34 So in the text says a container of hot liquid is placed in a freezer and it's kept at a constant temperature of 20 degrees fahrenheit, the initial temperature and so on.

00:46 So this is the constant temperature of the environment and the rate of change of the object of the temperature of the object is going to be proportional to it so the function y of t is going to be a solution of this differential equation.

01:09 Furthermore, we are given when t is zero that the initial temperature is 160 fahrenheit and we are given t is five that the temperature is in so t is given in minutes the temperature is 60 degrees all right what we want we want we want when t equals we want the time needed in minutes how many minutes are needed for the temperature to become 30 degrees okay so in order for us to find t we must find the y as a function of t which is the solution to this to this differential equation, all right? the first thing we do here is to separate the variables, d t, so we will multiply the d t to move it from the denominator here, we will divide with y minus 20.

02:20 So we will have d y over d t is equal to k now, d y over y minus 20 is going to be k times dt.

02:37 Once we have the variable separated we integrate both sides by each of the variable so this is going to be the natural logarithm of y minus 20 and that usually is in in absolute values and this is k t and instead of writing plus c p plus d, we write the one constant for both the left and the right hand side.

03:15 Y is obviously positive and it's a greater than 20, so we can lose these absolute values.

03:22 We apply the inverse function of the natural logarithm, which is the exponential function.

03:29 So we have y minus 20 is equal to e to the kt plus c1.

03:34 And this is a sum of exponents so it's by one of the first rules of exponents equal to the product of power so e to the c1 we will write this as a constant c which means that our general solution to the differential equation is y is equal to we will add 20 to both sides.

04:09 So y will be equal to c times e to the k t plus 20.

04:16 Okay, this is the general solution...

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