Turkey Cooking Problem - First Order

C

Calihan

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Dec 8, 2009
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Hi guys, first time here. Anyway I have this problem and I think I have a solution for it but I'm not sure I set up my original equation correct. Here is the problem:

A 16-lb turkey initially at 75 degrees is placed in an oven. After 75 minutes the turkey reaches 100 degrees. When will the turkey reach 160 degrees?

I am thinking that the rate at which the turkey cooks is directly proportional to the time its been in the oven, so I set up my equation as such: dy/dt = ky , with k being some constant.
So heres what I have done:

dy/dt = ky
1/y dy = k dt
integrate both sides to get ln|y| = kt + c1 (some constant)
y = e^(kt) * e^(c1)
y= B * e^(kt)

Using the intial conditions, I set the equation to 75 and solve for B first:
y(0) = 75 implies:

75 = B * e^(0*t)
75 = B * 1
B = 75

So then we can write the original equation as:

y = 75e^(kt)

Applying the next condition:
y(75) = 100
100 = 75e^(k * 75)
100/75 = e^(75k)
ln|100/75| = 75k
ln|100/75| / 75 = k
k = roughly .0051

So now we have the equation y = 75e^(.0051t)
And to find the answer to the question of when it will reach 160 degrees we set it equal to 160 as such:
160 = 75e^(.0051t)
160/75 = e^(.0051t)
ln|160/75| = .0051t
ln|160/75| / .0051 = t
t = about 148 minutes

That answer seems a bit off, so I am wondering if I set up my original equation wrong. Any suggestions, or am I just over thinking it?
 
Please define your "y".

and

What is the answer - how much is a "bit"-off?
 
I guess I could have used t as the variable, but in the problem I asked for how many mintues it takes to cook to a certain temp, and my answer was labeled in minutes.

But I figured it out, its just Newtons Lay of Cooling here, so I set it up like this:

dH/dt = -k(H - (whatever temp the oven is))

And go from there will give the answer.
So then my original equation is totally wrong because I didn't even use what temp the oven was at, silly me.
 
Calihan said:
I guess I could have used t as the variable, but in the problem I asked for how many mintues it takes to cook to a certain temp, and my answer was labeled in minutes.

But I figured it out, its just Newtons Lay of Cooling (in reverse - law of conductive heating) there, so I set it up like this:

dH/dt = -k(H - (whatever temp the oven is))

Still - you did not define your variables - what is "H"? It will be perfectly legitimate for your teacher to give a B - even if you got correct answer

And go from there will give the answer.
So then my original equation is totally wrong because I didn't even use what temp the oven was at, silly me.
Click to expand...
 
We can all used the variable, but in the problem I asked for how many minuets it takes to cook to a certain temp, and my answer was labeled in minutes.

sometimes mathematics it's not easy you take many long time to probe that your answer is correct math make you sense to think!

and by the way i am apologize for my bad English hope you understand it,and thanks for all your very good and helpful suggestion thanks for that againhow to hypnotize people
 
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