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sankalpmittal
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Homework Statement
A hot body placed in the surrounding of temperature θo obeys Newton's law of cooling dθ/dt = -k(θ-θo), where k is any constant. Its temperature is θ1 at t=0. t is time. The specific heat capacity of body is "s" and its mass is "m". Find the time starting from t=0 in which it will lose 90% of the maximum heat.
Homework Equations
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html
The Attempt at a Solution
I solved it via integration and got the correct answer as well. As far as I know, it can be solved without integration:
The maximum heat body can lose has to be ms(θ1-θo). If only 90 percent of this maximum heat is lost, we have, Heat lost = 9ms(θ1-θo)/10. As ms is constant we have, Temperature difference = 9(θ1-θo)/10
Initial temperature of body at time t=0 : θ1
Final temperature of body when it cools as per condition : θ1 - 9(θ1-θo)/10 = (θ1 + 9θo)/10
Average temperature of body = (Initial temperature + Finial temperature)/2 = (11θ1 + 9θo)/20
In magnitude only, we can write law as, Δθ/Δt= k(θ-θo)
Now change in temperature Δθ = Final temperature of body when it cools as per condition - Initial temperature of body at time t=0 = 9(θ1-θo)/10
Now Δt = t-0=t
θ-θo = Average temperature of body - θo = 11(θ1-θo)/20
Now putting all these values in Δθ/Δt= k(θ-θo), we get
9/t = 11k/2 => t = 18/11k
But the answer is t= ln(10)/k = 2.303/k.
Where did I go wrong ?
Please help !
Thanks in advance...