Finding the initial temperature using Newton's law
I am confused about how to solve the given problem :
A metal bar is heated 100c by a heat source. The heat source is removed when the temperature of the metal bar reached to a plateau. Now the metal bar is placed in a room. The room temperature is 25c. After 15 minutes the bar temperature reached to 90c. What was the initial temperature of the metal bar, assume newton's law of cooling applies.
Normally the initial temperature is given for other problems. it's easy to find out the value of the variables using T(t) = ce^kt+Ta formula. but here How to solve the problem?
calculus integration initial-value-problems
asked Nov 26 at 18:29
Cynthia
51
add a comment |
I am confused about how to solve the given problem :
A metal bar is heated 100c by a heat source. The heat source is removed when the temperature of the metal bar reached to a plateau. Now the metal bar is placed in a room. The room temperature is 25c. After 15 minutes the bar temperature reached to 90c. What was the initial temperature of the metal bar, assume newton's law of cooling applies.
Normally the initial temperature is given for other problems. it's easy to find out the value of the variables using T(t) = ce^kt+Ta formula. but here How to solve the problem?
calculus integration initial-value-problems
asked Nov 26 at 18:29
Cynthia
51
add a comment |
I am confused about how to solve the given problem :
A metal bar is heated 100c by a heat source. The heat source is removed when the temperature of the metal bar reached to a plateau. Now the metal bar is placed in a room. The room temperature is 25c. After 15 minutes the bar temperature reached to 90c. What was the initial temperature of the metal bar, assume newton's law of cooling applies.
Normally the initial temperature is given for other problems. it's easy to find out the value of the variables using T(t) = ce^kt+Ta formula. but here How to solve the problem?
calculus integration initial-value-problems
asked Nov 26 at 18:29
Cynthia
51
I am confused about how to solve the given problem :
A metal bar is heated 100c by a heat source. The heat source is removed when the temperature of the metal bar reached to a plateau. Now the metal bar is placed in a room. The room temperature is 25c. After 15 minutes the bar temperature reached to 90c. What was the initial temperature of the metal bar, assume newton's law of cooling applies.
Normally the initial temperature is given for other problems. it's easy to find out the value of the variables using T(t) = ce^kt+Ta formula. but here How to solve the problem?
calculus integration initial-value-problems
calculus integration initial-value-problems
asked Nov 26 at 18:29
Cynthia
51
asked Nov 26 at 18:29
Cynthia
51
asked Nov 26 at 18:29
Cynthia
51
asked Nov 26 at 18:29
Cynthia
51
asked Nov 26 at 18:29
Cynthia
51
51
add a comment |
add a comment |
1 Answer
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Here you have two sets of data: T(0)=100 and T(15)=90 while $T_a=25$. So you have two equations:$$100=c*e^{k*0}+25 (1)$$$$90=c*e^{15k}+25 (2)$$
From (1) find c, from (2) find k. Now you have equation in the form $$T=c*e^{k*t}+T_a$$
with c and k known. Here $T_a$ = 100, because "plateau" happens when $T=T_a = 100$. Maybe this problem lacks info about time of heating. If you'll have time of heating, you'll easily solve it.
answered Nov 26 at 19:47
Kelly Shepphard
2298
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
thank you so much
– Cynthia
Dec 1 at 18:13
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
|
show 2 more comments
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1 Answer
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1 Answer
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Here you have two sets of data: T(0)=100 and T(15)=90 while $T_a=25$. So you have two equations:$$100=c*e^{k*0}+25 (1)$$$$90=c*e^{15k}+25 (2)$$
From (1) find c, from (2) find k. Now you have equation in the form $$T=c*e^{k*t}+T_a$$
with c and k known. Here $T_a$ = 100, because "plateau" happens when $T=T_a = 100$. Maybe this problem lacks info about time of heating. If you'll have time of heating, you'll easily solve it.
answered Nov 26 at 19:47
Kelly Shepphard
2298
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
thank you so much
– Cynthia
Dec 1 at 18:13
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
|
show 2 more comments
Here you have two sets of data: T(0)=100 and T(15)=90 while $T_a=25$. So you have two equations:$$100=c*e^{k*0}+25 (1)$$$$90=c*e^{15k}+25 (2)$$
From (1) find c, from (2) find k. Now you have equation in the form $$T=c*e^{k*t}+T_a$$
with c and k known. Here $T_a$ = 100, because "plateau" happens when $T=T_a = 100$. Maybe this problem lacks info about time of heating. If you'll have time of heating, you'll easily solve it.
answered Nov 26 at 19:47
Kelly Shepphard
2298
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
thank you so much
– Cynthia
Dec 1 at 18:13
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
|
show 2 more comments
Here you have two sets of data: T(0)=100 and T(15)=90 while $T_a=25$. So you have two equations:$$100=c*e^{k*0}+25 (1)$$$$90=c*e^{15k}+25 (2)$$
From (1) find c, from (2) find k. Now you have equation in the form $$T=c*e^{k*t}+T_a$$
with c and k known. Here $T_a$ = 100, because "plateau" happens when $T=T_a = 100$. Maybe this problem lacks info about time of heating. If you'll have time of heating, you'll easily solve it.
answered Nov 26 at 19:47
Kelly Shepphard
2298
Here you have two sets of data: T(0)=100 and T(15)=90 while $T_a=25$. So you have two equations:$$100=c*e^{k*0}+25 (1)$$$$90=c*e^{15k}+25 (2)$$
From (1) find c, from (2) find k. Now you have equation in the form $$T=c*e^{k*t}+T_a$$
with c and k known. Here $T_a$ = 100, because "plateau" happens when $T=T_a = 100$. Maybe this problem lacks info about time of heating. If you'll have time of heating, you'll easily solve it.
answered Nov 26 at 19:47
Kelly Shepphard
2298
answered Nov 26 at 19:47
Kelly Shepphard
2298
answered Nov 26 at 19:47
Kelly Shepphard
2298
answered Nov 26 at 19:47
Kelly Shepphard
2298
2298
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
thank you so much
– Cynthia
Dec 1 at 18:13
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
|
show 2 more comments
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
thank you so much
– Cynthia
Dec 1 at 18:13
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
Thanks for the reply, I was also thinking the same that T(0) =100 but here I have to find out the initial temp. of the metal bar which is also T(0). That is actually confusing. after find out C and K as you solved how to find the initial temperature? Can you please explain how the time of heating can solve the problem easily?
– Cynthia
Nov 26 at 20:44
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
T(0) = 100 for cooling process at room temperature, but T(0) is unknown for heating (two different T(0)). You can name them $T_c (0) = 100$ and $T_h (0) = ?$, if it's still confusing. About solution: since you know c,k, and $T_a$ (note: you should use -c, not +c, because not it's not cooling; it's heating - opposite process), last equation has all needed unknowns except t (time of heating). Just solve it for T. This T is the answer. $$T=c*e^{kt}+T_a$$
– Kelly Shepphard
Nov 27 at 10:38
thank you so much
– Cynthia
Dec 1 at 18:13
thank you so much
– Cynthia
Dec 1 at 18:13
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
No problem :) I'm glad that this info was useful for you.
– Kelly Shepphard
Dec 1 at 18:25
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
Sorry I just need to clear one thing. you said we should use -c as it is heating. so the last equation should be : T=-c∗e^kt+Ta. right?
– Cynthia
Dec 1 at 19:12
|
show 2 more comments
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