A)
∫dTT−Ta=−∫k dt\int \cfrac{dT}{T-T_a} = - \int k\,dt∫T−TadT=−∫kdt
ln ∣T−Ta∣=−kt+ln C0ln\,|T - T_a| = -kt + ln\,C_0ln∣T−Ta∣=−kt+lnC0
T−Ta=e−kt+ln C0T - T_a = e^{-kt + ln\,C_0}T−Ta=e−kt+lnC0
T−Ta=eln C0 e−ktT - T_a = e^{ln\,C_0}\,e^{-kt}T−Ta=elnC0e−kt
T−Ta=C0 e−ktT - T_a = C_0\,e^{-kt}T−Ta=C0e−kt
B)
70−20=C0 e−k∗0C0=50‾70 - 20 = C_0\,e^{-k*0} \\ \underline {C_0 = 50}70−20=C0e−k∗0C0=50
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Dear Teresa Judge-Bird, in the integral of k=k*t ^0 the formulae ∫kf(t)dt=k ∫f(t)dt and ∫t^0dt=t^(0+1)/(0+1)+C1 were applied. The integration constant C1 was denoted by ln C_0 in a solution of the question.
Hi I am struggling to understand how you went from ∫kdt to kt+lnC0?
The variables T and t were separated, the corresponding left-hand and right-hand sides of a new differential equation were integrated. Next, the logarithmic equation was solved. The absolute value sign can be omitted using the sign of C_0 in a solution of the question.
can you explain more how did you answer part a?
Comments
Dear Teresa Judge-Bird, in the integral of k=k*t ^0 the formulae ∫kf(t)dt=k ∫f(t)dt and ∫t^0dt=t^(0+1)/(0+1)+C1 were applied. The integration constant C1 was denoted by ln C_0 in a solution of the question.
Hi I am struggling to understand how you went from ∫kdt to kt+lnC0?
The variables T and t were separated, the corresponding left-hand and right-hand sides of a new differential equation were integrated. Next, the logarithmic equation was solved. The absolute value sign can be omitted using the sign of C_0 in a solution of the question.
can you explain more how did you answer part a?