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Calculus IChapter 1: ReviewSection 1-7

Exponential Functions — Question 6

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Question 6

A forest fire is spreading in a circular pattern and the area (in square kilometers) covered by the fire after tt hours is modeled by the function: A(t)=2.5e0.4tA(t) = 2.5e^{0.4t}

  • (a) What is the area affected after 6 hours?

  • (b) After how many hours will the fire have covered 50 square kilometers?

  • (c) How fast is the area growing (in km2^2/hr) after 6 hours?

Round all answers to two decimal places.

Original worksheet page 1: question and worked solution for 1-7-006
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Question 6 - Solution

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

(a)

A(6)=2.5e2.4A(6)=2.5e^{2.4}

27.56km2\boxed{27.56\quad\mathrm{km}^2}

(b)

t=ln⁡(50/2.5)/0.4t=\ln(50/2.5)/0.4

7.49hours\boxed{7.49\quad\text{hours}}

(c)

A′(t)=e0.4t,A′(6)=e2.4A'(t)=e^{0.4t},\quad A'(6)=e^{2.4}

11.02km2/hr\boxed{11.02\quad\mathrm{km}^2/\mathrm{hr}}

Original worksheet page 2: question and worked solution for 1-7-006

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