According to Newton’s law of cooling, the temperature T of

Chapter 3, Problem 11E

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QUESTION:

According to Newton’s law of cooling, the temperature T of a body at time t is given by \(T=T_{a}+\left(T_{0}-T_{a}\right) e^{-k t}\), where \(T_{a}\) is the ambient temperature, \(T_0\) is the initial temperature, and k is the cooling rate constant. For a certain type of beverage container, the value of k is known to be \(0.025 \min ^{-1}\).

a. Assume that \(T_{a}=36^{\circ} \mathrm{F}\) exactly and that \(T_{0}=72.0 \pm 0.5^{\circ} \mathrm{F}\). Estimate the temperature T at time \(t=10 \min\), and find the uncertainty in the estimate.

b. Assume that \(T_{0}=72^{\circ} \mathrm{F}\) exactly and that \(T_{a}=36.0 \pm 0.5^{\circ} \mathrm{F}\). Estimate the temperature T at time \(t=10 \mathrm{~min}\), and find the uncertainty in the estimate.

Equation Transcription:

Text Transcription:

T=T_a+(T_0-T_a)e^{-kt}

T_a

T_0

0.025 min^-1

T_a=36^oF

T_0=72.0{+/-}0.5^oF

t=10 min

T_0=72^oF

T_a=36.0{+/-}0.5^oF

t=10 min

Questions & Answers

QUESTION:

According to Newton’s law of cooling, the temperature T of a body at time t is given by \(T=T_{a}+\left(T_{0}-T_{a}\right) e^{-k t}\), where \(T_{a}\) is the ambient temperature, \(T_0\) is the initial temperature, and k is the cooling rate constant. For a certain type of beverage container, the value of k is known to be \(0.025 \min ^{-1}\).

a. Assume that \(T_{a}=36^{\circ} \mathrm{F}\) exactly and that \(T_{0}=72.0 \pm 0.5^{\circ} \mathrm{F}\). Estimate the temperature T at time \(t=10 \min\), and find the uncertainty in the estimate.

b. Assume that \(T_{0}=72^{\circ} \mathrm{F}\) exactly and that \(T_{a}=36.0 \pm 0.5^{\circ} \mathrm{F}\). Estimate the temperature T at time \(t=10 \mathrm{~min}\), and find the uncertainty in the estimate.

Equation Transcription:

Text Transcription:

T=T_a+(T_0-T_a)e^{-kt}

T_a

T_0

0.025 min^-1

T_a=36^oF

T_0=72.0{+/-}0.5^oF

t=10 min

T_0=72^oF

T_a=36.0{+/-}0.5^oF

t=10 min

ANSWER:

Answer :

Step 1 of 3:

        According to the Newton’s law of cooling, the temperature T of a body at time t is given by  

                   

                   Where,  is the ambient temperature

                                   is the initial temperature

                                    k is the cooling rate constant.

                       Here the value of k is 0.025


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