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Calculus IChapter 1: ReviewSection 1-7

Exponential Functions — Question 2

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Question 2

A radioactive substance decays according to the formula A(t)=A0e−ktA(t) = A_0 e^{-kt} where A0=100A_0 = 100 grams and the half-life of the substance is 10 hours.

  • (a) Find the decay constant kk.

  • (b) How much of the substance remains after 24 hours?

  • (c) After how many hours will only 20 grams remain?

Round all answers to two decimal places.

Original worksheet page 1: question and worked solution for 1-7-002
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Question 2 - Solution

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

(a)

e−10k=1/2⇒k=ln⁡(2)/10e^{-10k}=1/2\ \Longrightarrow\ k=\ln(2)/10

0.07hr−1\boxed{0.07\quad\mathrm{hr}^{-1}}

The exact value of kk above is used in later parts; its decimal value is 0.0693150.069315.

(b)

A(24)=100e−24ln⁡(2)/10A(24)=100e^{-24\ln(2)/10}

18.95grams\boxed{18.95\quad\text{grams}}

(c)

100e−kt=20⇒t=10ln⁡(5)/ln⁡(2)100e^{-kt}=20\ \Longrightarrow\ t=10\ln(5)/\ln(2)

23.22hours\boxed{23.22\quad\text{hours}}

Original worksheet page 2: question and worked solution for 1-7-002

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