Question #122484
The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (2), given in Section 3.1, is dT/dt = kS(T-Tm), where
k < 0
and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 65° C. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 38° C. If Tm = 21° C, then what is the temperature of the coffee in cup B after 30 min? (Round your answer to two decimal places.)

______ degrees Celsius
1
Expert's answer
2020-06-23T15:16:15-0400

dTdt\frac {dT}{dt} = kS(T-Tm) where k < 0

Here Tm= 21°C and S is exposed surface area.

Here SBSA=2\frac {S_B}{S_A} = 2 , where SA and SB are the exposed surface area of A-cup and B-cup respectively.

Therefore

dT(T21)\frac {dT}{(T - 21)} = kSdt

Integrating,

\int dT(T21)\frac {dT}{(T - 21)} = \int kS dt

=> ln(T-21) = kSt + C, C is integration constant.

Temperature are being measured in °C and time in minute here.

When t = 0 , initial temperature T = 65

So, ln(65 - 21) = C

=> C = ln(44)

So ln(T-21) = kSt + ln(44 )

So ln(T-21) - ln(44) = kSt

=> ln T2144\frac {T-21}{44} = kSt

For cup A, S = SA and T = 38 when t = 30

So equation for cup-A is

ln 382144\frac {38-21}{44} = 30kSA

=> ln 1744\frac {17}{44} = 30kSA •••••••••eq(1)

For cup-B, S = SB and T=? when t = 30

So equation for cup-B is

ln T2144\frac {T-21}{44} = 30kSB

ln T2144\frac {T-21}{44} = 60kSA as SB =2 SA ••••eq(2)

Comparing eq(2) and eq(1) we get

ln T2144\frac {T-21}{44} = 2 ln 1744\frac {17}{44}

=> ln T2144\frac {T-21}{44} = ln (1744\frac {17}{44} )2

=> T2144\frac {T-21}{44} = (1744)2(\frac {17}{44})^2

=> T - 21 = 28944\frac {289}{44}

=> T = 21 + 6.568

=> T = 27.568

The temperature of coffee in cup B is 27.57 °C ( correct up to two decimal places)





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