Solve Temperature Question: Find a, b & k

[Burjam]

Homework Statement

The temperature, T, in ºC, of a yam put into a 200ºC oven is given as a function of time, t, in minutes, by

T=a(1-e^-kt)+b

a. If the yam starts at 20ºC, find a and b.
b. If the temperature of the yam is initially increasing at 2ºC per minute, find k.

Homework Equations

Given in problem.

The Attempt at a Solution

I'm not really sure how to set up this problem to do part a. I think if I had a and b I'd be able to do part b with some derivatives and algebra but I just can't figure out how to get there. My instincts are telling me to set T = 20 = a(1-e^-kt)+b but I don't know how to approach this from here.

 

[Qube]

For part a set time = 0. What happens to e^-kt then? Can you solve for one of the constants?

 

[Burjam]

When t=0, e=1 and therefore a=0. Therefore b=initial T which in the case of a is 20. Ok great. But what do I do to get a? I think I need to set T=20 but I don't know what I need to set t equal to. I know I can't use zero, but aside from that I don't know what to do.

 

[Qube]

When t=0, e=1 and therefore a=0.

Generalizing your statement, you're saying that if any number x is multiplied by zero, that number x is zero.

Do you see a problem with this?

If you have x*0, does this imply that x = 0?

When t=0, e=1 and therefore a=0.

Also, is e a constant or a variable?

 

[Burjam]

Generalizing your statement, you're saying that if any number x is multiplied by zero, that number x is zero.

Do you see a problem with this?

I'm saying that at t=0, a(1-e^-kt) will equal zero because e⁰=1, 1-1=0 and a*0=0.

If you have x*0, does this imply that x = 0?

I wasn't trying to imply that.

Also, is e a constant or a variable?

e is the constant Euler's number.

 

[Qube]

I'm saying that at t=0, a(1-e^-kt) will equal zero because e⁰=1, 1-1=0 and a*0=0.

a*0 indeed equals 0, but we're deviating from the original problem. Are there other parts to the original equation even when you substitute 0 in for time and 20 degrees C in for the temperature?

e is the constant Euler's number.

Right, and e = 2.71 (approximately). Not 1.

 

[Burjam]

a*0 indeed equals 0, but we're deviating from the original problem. Are there other parts to the original equation even when you substitute 0 in for time and 20 degrees C in for the temperature?

What do you mean by "other parts"?

Right, and e = 2.71 (approximately). Not 1.

But if we're setting t=0, it will be e^-k*0. -k*0=0 so it will be e⁰=1.

 

[Qube]

What do you mean by "other parts"?



But if we're setting t=0, it will be e^-k*0. -k*0=0 so it will be e⁰=1.

What happened to the variables you were solving for? Surely if they all canceled out then this would be a bad question on the part of the teacher, right?

e^0 = 1.

You stated e = 1.

 

[Burjam]

What happened to the variables you were solving for? Surely if they all canceled out then this would be a bad question on the part of the teacher, right?

Wait what? What do you mean "what happened to the variables you were solving for?" I was only trying to say everything but b is canceled out when t=0. Surely you must see that.

e^0 = 1.

You stated e = 1.

Ok that was a false. Take it as e⁰=1

 

[Dick]

Wait what? What do you mean "what happened to the variables you were solving for?" I was only trying to say everything but b is canceled out when t=0. Surely you must see that.



Ok that was a false. Take it as e⁰=1

Ok. So you've got b. Now what happens if t is very, very large?

 

[Qube]

Wait what? What do you mean "what happened to the variables you were solving for?" I was only trying to say everything but b is canceled out when t=0. Surely you must see that.

T = a(y) + b, where y is substituted in for whatever function that a was multiplied by.

If y = 0, then

T = a(0) + b

Do you see what happens (contrary to what you think?)

 

[Burjam]

Ok. So you've got b. Now what happens if t is very, very large?

When t gets very large, e^-kt approaches 0. So the equation becomes a(1-0)+b=T. This is simplified to a+b=T where b is the initial temperature and T is the final temperature. Therefore a=T-b.

T = a(y) + b, where y is substituted in for whatever function that a was multiplied by.

If y = 0, then

T = a(0) + b

Do you see what happens (contrary to what you think?)

Yes I get it. But in this scenario a(0)=0. I just simplified it.

 

[Dick]

When t gets very large, e^-kt approaches 0. So the equation becomes a(1-0)+b=T. This is simplified to a+b=T where b is the initial temperature and T is the final temperature. Therefore a=T-b.

Yes, that's it. t=0 gives you one constant and t=infinity gives you the other one. Now you just need to find k. You are given dT/dt=2 at t=0. So?

 

[Burjam]

Solving for k I get 1/90, which appears to be the right answer in the back of the book. Along with my answers for a and b. Thanks.