For the following exercises, use this scenario: A pot of...

For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of $100^{\circ}$ Fahrenheit was taken off the stove to cool in a $69^{\circ} \mathrm{F}$ room. After fifteen minutes, the internal temperature of the soup was $95^{\circ} \mathrm{F}$. Use Newton's Law of Cooling to write a formula that models this situation.

For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of $100^{\circ}$ Fahrenheit was taken off the stove to cool in a $69^{\circ} \mathrm{F}$ room. After fifteen minutes, the internal temperature of the soup was $95^{\circ} \mathrm{F}$.
Use Newton's Law of Cooling to write a formula that models this situation.

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling describes how the temperature of an object changes in relation to the temperature of its environment. The rate at which the object cools (or warms) is proportional to the difference between its temperature and the ambient temperature, leading to an exponential decay model.

Exponential Decay Model

The exponential decay model characterizes the process by which the temperature difference between an object and its environment decreases over time. This model is mathematically represented through an equation where the future temperature depends on the initial temperature, the ambient temperature, and a decay constant.

Differential Equations

Differential equations are used to formalize the relationship described by Newton's Law of Cooling. Specifically, the cooling process is modeled by a first-order differential equation that equates the time derivative of the object's temperature to a constant multiplied by the difference between the object's temperature and that of the surroundings.

Initial Conditions and Parameter Estimation

To accurately model a cooling process, initial conditions such as the initial temperature of the object and the ambient temperature must be known. Additionally, information about the temperature of the object after a certain period is used to determine the cooling constant, which is an essential parameter in the exponential decay function.

Transcript

00:01 So for this problem, we have that a pot of boiling soup with an internal temperature of 100 degrees fahrenheit was taken off the stove to cool in a 69 degree fahrenheit room.

00:17 After 15 minutes, the internal temperature of the soup was 95 degrees fahrenheit.

00:26 We are going to use newton's law of cooling to write a form.

00:30 Then models this situation.

00:33 So we're going to start off with newton's law of cooling, which is t of t, which stands for time.

00:42 I'm going to highlight t as for time.

00:45 And then a here is the difference between the initial temperature and the surrounding temperature times e to the k times t, and then k is the cooling rate of the continuous rate of cooling, sorry, plus t sub s and t sub s is the surrounding temperature.

01:11 So let's go ahead and start the problem.

01:17 So we're going to start with the fact that at our initial temperature, we have 100 degrees fahrenheit.

01:27 And so let's plug this in into our formula and see what we can do.

01:32 What we can solve for.

01:34 So we'll have 100 equals a.

01:39 We don't know what a is at this point.

01:42 E.

01:43 K times t, but t, notice t is zero.

01:50 Plus our surrounding temperature is 69.

01:54 So now we can go ahead and solve for a.

01:59 So let's see.

02:01 We are going to subtract 69 from both sides, so i'm just going to skip writing that step.