Tricky calculus question: rate of change of temperature

[Guest]

When the temperature of an object is To, the rate of change of T is degrees / minute, where R = k(T - 16), k being a constant.

a. Prove that the function T = 16 + Ae^(kt), where A is a constant and t the time in minute satisfies this condition.
b. If, initially T = 0, and after 10 minutes, T = 12, find the values of A and e^(10k).
c. Find the temperature of the object after a further 5 minutes.
d. Sketch the graph of T, indicating its behaviour as t continues to increase.

What I intend to do:

For a) I am not sure but I was thinking of actually substituting To into the R = ....equation and then see if they are equal?
b) not sure about this one. I will think harder and try to post my working out later
c)?
d)I need answers to b & c.

thank you in advance...

 

[skeeter]

nothing "tricky" here ... just a basic exponential function defined by its given rate of change

dT/dt = k(T - 16)

separate variables ...

dT/(T - 16) = k dt

integrate ...

ln|T - 16| = kt + C

convert to an exponential function ...

T - 16 = e^{kt + C}

use properties of exponents ...

T - 16 = e^Ce^kt

let A = e^C, a constant ...

T = 16 + Ae^kt

can you finish the rest of the problem?

 

[Guest]

I surely can...thank you

for your help...