Logarithmic Cooling Model Analysis

00:00 7 marks

An engineer is analyzing the cooling of a heated object using Newton’s law of cooling, where the temperature model is given by \(T(t)=T_A+(T_0-T_A)e^{-kt}\). For a specific experiment, the initial temperature is \(T_0=200^\circ C\), the ambient temperature is \(T_A=25^\circ C\), and the object cools to \(100^\circ C\) in 5 minutes.

(a) Set up the logarithmic equation to solve for the cooling constant \(k\).

(b) Using the value of \(k\) from part (a), determine the time \(t\) when the temperature reaches \(50^\circ C\).

Part a

4 minutes 3 marks

Question

Substitute \(t=5\) into the model \(T(t)=T_A+(T_0-T_A)e^{-kt}\) to form the equation \(100=25+175e^{-5k}\). Solve this equation to express \(k\) in terms of logarithms.

Hint Hint

Hint

Isolate the exponential term and take the natural logarithm of both sides.

Answer Answer
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Answer Answer

Complete Answer

\(k=\frac{1}{5}\ln\left(\frac{7}{3}\right)\)

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Part b

6 minutes 4 marks

Question

Set \(T(t)=50\) in the model \(T(t)=25+175e^{-kt}\) and solve for the time \(t\) using the value of \(k\) from part (a).

Hint Hint

Hint

Express the equation so that \(e^{-kt}\) is isolated and take logarithms to solve for \(t\).

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

\(t=\frac{5\ln(7)}{\ln(7/3)}\)

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