Question #122480
1. A thermometer reading 20° C is placed in an oven preheated to a constant temperature. Through a glass window in the oven door, an observer records that the thermometer reads 42° C after
1/2 minute and 62° C after 1 minute. How hot is the oven?

_____ degrees Celsius
1
Expert's answer
2020-06-22T18:15:25-0400

Let T represents temperature of thermometer and S be the constant temperature of oven.

t represents time

Here all temperatures are in °C and time in minute

dTdt\frac {dT}{dt} is proportional to differenc of temperature of thermometer and that of oven.

So dTdt\frac {dT}{dt} = k(T-S), k is constant

=> dTTS\frac {dT}{T-S} = k dt

Integrating

\int dTTS\frac {dT}{T-S} = \int k dt

ln | T-S | = kt + C , C is integration constant

=> ln(S-T) = kt + C as S > T here

When t = 0 , T = 20

So ln(S-20) = k*0 + C

=> C = ln(S-20)

So ln(S-T) = kt + ln(S-20)

=> ln(S-T) - ln(S-20) = kt

=> ln STS20\frac {S-T}{S-20} = kt

Two set of values of t, T are as follows

When t = 12\frac {1}{2} , T = 42

and when t = 1, T = 62

So ln S42S20\frac {S-42}{S-20} = k2\frac {k}{2} ..........eq(1)

and ln S62S20\frac {S-62}{S-20} = k ............eq(2)

Comparing eq(1) and eq(2)

2 ln S42S20\frac {S-42}{S-20} = ln S62S20\frac {S-62}{S-20}

=> ln [S42S20]2[\frac {S-42}{S-20} ]^2 = ln S62S20\frac {S-62}{S-20}

=> [S42S20]2[\frac {S-42}{S-20} ]^2 = S62S20\frac {S-62}{S-20}

=> (S42)2(S20)\frac {(S-42)^2}{(S-20)} = (S-62) as S>20, (S - 20)≠0

=> (S-42)2 = (S-20)(S-62)

=> S2 - 84S + 1764 = S2 - 82S + 1240

=> - 84S + 1764 = - 82S + 1240

=> -84S + 82S = 1240 - 1764

=> -2S = -524

=> S = 524/2 = 262

The oven is 262 degree Celsius



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