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Calculus IChapter 1: ReviewSection 1-7

Exponential Functions — Question 3

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Question 3

A cup of coffee is initially at 190°F and is left to cool in a room that is maintained at 70°F. After 15 minutes, the coffee has cooled to 130°F.

Assume the temperature of the coffee follows Newton’s Law of Cooling: T(t)=Troom+(T0−Troom)e−ktT(t) = T_{\text{room}} + (T_0 - T_{\text{room}})e^{-kt}

  • (a) Find the value of the constant kk.

  • (b) What will the temperature be after 30 minutes?

  • (c) After how many minutes will the coffee reach 100°F?

Round all answers to two decimal places.

Original worksheet page 1: question and worked solution for 1-7-003
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Question 3 - Solution

Use the exact exponential model and keep full precision until the final rounding.

(a)

130=70+120e−15k⇒k=ln⁡(2)/15130=70+120e^{-15k}\ \Longrightarrow\ k=\ln(2)/15

0.05min−1\boxed{0.05\quad\mathrm{min}^{-1}}

The exact value of kk above is used in later parts; its decimal value is 0.0462100.046210.

(b)

T(30)=70+120e−2ln⁡2T(30)=70+120e^{-2\ln2}

100.00∘F\boxed{100.00\quad{}^\circ\mathrm F}

(c)

100=70+120e−kt⇒t=ln⁡(4)/k100=70+120e^{-kt}\ \Longrightarrow\ t=\ln(4)/k

30.00minutes\boxed{30.00\quad\text{minutes}}

Original worksheet page 2: question and worked solution for 1-7-003

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