The value of D and a in T(t)=T(0)+Dat. To estimate the temperature of

EunoR

EunoR

Answered question

2021-09-25

The value of D and a in T(t)=T(0)+Dat.
To estimate the temperature of pan after 10 minutes.
The time taken by pan to reach 400 F.

Answer & Explanation

SoosteethicU

SoosteethicU

Skilled2021-09-26Added 102 answers

Given:
T(t)=T(0)+Dat and T0 is 320 F. Boiling water with a temperature of 2120 and it cools to 700 in 30 minutes. Here D is the difference between room temperature and water temperature.
Calculation:
a) To determine the value of D and a:
We have T(t)=T(0)+Dat ----(1).
Here T0=320F ----(2)
Water temperature is 2120F
Hence D=21232=180,
Next, D=180,T0=32 and t=30 minutes and t=12 hour.
Substitute the above values in the equation (1),
T(12)=32+180a12
Here T(12)=70. On substituting the value of T(12) in the above we get,
70=32+180a127032=180a12180a12=38a12=38180a12=0.211(a12)2=(0.211)2a=0.044.
Hence the values of D, T0 and a are 180, 32 and 0.044 respectively.
b) To estimate the temperature of pan after 10 minutes.
Here 10 minutes =16 hour. So, t=16.
Substitute the value of t, a, D and t0 in the equation (1) we get,
T(12)=32+180(0.044)16T(12)=32+180(0.594)T(12)=32+106.95T(12)=138.95T(12)=1390F
Hence the value of T(t) after 10 minutes is 1390F.
c) The time taken by pan to reach 400 F.
Given: T(t)=40,T0=32,D=180 and a=0.044.
Substitute the above values in equation (1) we get,
40=32+180(0.044)t4032=180(0.044)t180(0.044)t=8(0.044)t=8180(0.044)t=0.044
Applying log on both sides,
log(0.044)t=log0.044tlog0.044=log0.044t=1
Hence the time taken by pan to reach 400 F is 1 hour.
Conclusion:
The value of D is 180, a is 0.044 and T0 is 320 F. The temperature of the pan after 10 minutes is 1390 F. The time taken by pan to reach 400 F is 1 hour.

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