Solve 1st-Order ODE: Chicken in 375° Oven

In summary, the conversation discusses using Newton's law of cooling to determine the time it takes for a 3lb chicken to reach a temperature of 150 degrees in a 375 degree oven. The conversation also mentions using initial conditions and solving for k and t to find the desired time.
  • #1
find_the_fun
148
0
I'm going to need a little help with this one. I get an answer but it doesn't make sense. The question states

According to Newton's law of cooling, the time rate of change of temperature T(t) of a body immersed in a medium of constant temperature A is proportional to the difference A-T. That is \(\displaystyle \frac{dT}{dt}=k(A-T)\) where k is a positive constant. A 3lb chicken is initially 50 degrees, is put into a 375 deg oven. After 75 minutes it is found that the chicken is 125 deg. When will it be 150 deg?

I may have gone astray right away. Are we trying to solve the differential equation and find T(t)? I found it to be \(\displaystyle T=A+\frac{C}{e^{kt}}\) and then using initial condition T(0)=50 found \(\displaystyle T=A+\frac{50-A}{e^{kt}}\) Then we know T(75)=125. This let's us solve for k. Hold on a sec, :confused: basically I just used one IC to solve for C and another to solve for K. I'm not sure that's right.
 
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  • #2
We really don't need to find $T$, what I would do to answer this question is separate variables, use the initial conditions and solve for $kt$, then use the other known point on the curve to solve for $k$, and then you will have $t$ as a function of $T$.
 

FAQ: Solve 1st-Order ODE: Chicken in 375° Oven

1. What is a first-order ODE?

A first-order ordinary differential equation (ODE) is a type of mathematical equation that describes the relationship between a function and its derivative. It involves only the first derivative of the function, hence the name "first-order."

2. How do you solve a first-order ODE?

To solve a first-order ODE, you need to find the function that satisfies the equation. This can be done using various methods, such as separation of variables, integrating factors, or the method of undetermined coefficients.

3. What does it mean to have a chicken in a 375° oven?

In this context, the chicken represents a physical object that is being heated in an oven at a temperature of 375° Fahrenheit. The temperature of the oven is a variable that can be used to model the rate of change of the chicken's temperature over time.

4. How is a chicken in a 375° oven related to a first-order ODE?

A chicken in a 375° oven can be used as an example to illustrate the concept of a first-order ODE. The rate at which the chicken's temperature changes can be described by a first-order ODE, where the oven temperature is the input and the chicken's temperature is the output.

5. What are some practical applications of solving first-order ODEs?

First-order ODEs are used in various fields of science and engineering to model and understand real-world phenomena, such as population growth, chemical reactions, and electrical circuits. They are also essential in the development of mathematical models for predicting and controlling various processes and systems.

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