Is My Cooling Equation Calculation Correct or Is There a Typo in My Textbook?

[xsgx]

I just need someone to confirm my answer so I can be certain that my book has a typo and I am not making a some minuscule mistake that is leading me to the wrong answer.

A piece of metal is heated to 300 degrees Celsius and then placed in a cooling liquid at 50 degrees Celsius. After 4 minutes, the metal has cooled to 175 degrees Celsius. Find it's temperature after 12 minutes.Equation: F(t)= To+ Ce^-kt

e= Euler's number

I started out solving for k

175= 50 + 300e^-k4
-50 -50

125=300e^-k4
/300 /300
-The Celsius units cancel out-

125/300=e^-k4
Ln Ln

Ln(125/300)= -4k
/-4 /-4

k= .21887

then I plug in the value of k and solve for what temperature the metal will be after 12 minutes have passed.

F(t)= 50+300e^-.21887*12

= 71.700

I end up getting the answer above but the book says the answer is 81.25

Thanks in advance for your help.

 

[haruspex]

I just need someone to confirm my answer so I can be certain that my book has a typo and I am not making a some minuscule mistake that is leading me to the wrong answer.

A piece of metal is heated to 300 degrees Celsius and then placed in a cooling liquid at 50 degrees Celsius. After 4 minutes, the metal has cooled to 175 degrees Celsius. Find it's temperature after 12 minutes.Equation: F(t)= To+ Ce^-kt

e= Euler's number

I started out solving for k

175= 50 + 300e^-k4
-50 -50

125=300e^-k4
/300 /300
-The Celsius units cancel out-

125/300=e^-k4
Ln Ln

Ln(125/300)= -4k
/-4 /-4

k= .21887

then I plug in the value of k and solve for what temperature the metal will be after 12 minutes have passed.

F(t)= 50+300e^-.21887*12

= 71.700

I end up getting the answer above but the book says the answer is 81.25

Thanks in advance for your help.

If you plug t=0 into your first equation, does it give the right initial temperature?

 

[xsgx]

If you plug t=0 into your first equation, does it give the right initial temperature?

Wow.. I can't believe I missed that. Thank you.

 

[haruspex]

Wow.. I can't believe I missed that. Thank you.

Btw, there's a very quick way to the answer in this case. You're given an initial temperature difference of 250C, and a later difference of 125C, so you know how long it takes for the difference to halve. In a further twice that time, it quarters to 31C.