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How Fast Is a Forest Fire Spreading? Exploring Related Rates in Calculus

A circular patch of forest is on fire, with a radius of 80 meters and spreading at a rate of 1000 square meters per minute. This scenario raises the question: how quickly is the radius of the blaze growing? To find this rate, we start by organizing our information and identifying known values, then write the relevant equation relating the area and radius. By differentiating with respect to time using the Chain Rule, we can solve for the change in the radius over time. Ultimately, we find that the radius increases at approximately 1.989 meters per minute.

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How Fast Is a Forest Fire Spreading? Exploring Related Rates in Calculus

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  1. Section 4.6Related Rates

  2. Spreading Like Wildfire! A circular patch of forest is en fuego. When the radius is 80m, the fire spreads at a rate of 1000 square meters / minute. How quickly is the radius of the blaze growing at that moment?

  3. 1. Organize Information • Identify what you know • r = 80 • dA/dt = 1000 (notice this is a rate of change) • Identify & Label what you’re trying to find • dr/dt = ??? • We will always be solving for another (related) rate

  4. 2. Write an equation • Must relate all variables whose rates we know or want to find • No derivatives yet!!! • No substituting yet!!!

  5. 3. Differentiate • Always differentiate with respect to time, t. • Remember to use the Chain Rule!! • dA/dt = 2πr dr/dt

  6. Solve • Plug in the numbers you know • dA/dt = 2π r dr/dt • 1000 = 2π (80) dr/dt • Solve for the unknown rate • Dr/dt = 1000/(160 π) ≈ 1.989 meters / min • The radius of the forest fire is increasing at 1.989 meters per minute. • Remember to include proper units

  7. Summary • Organize information • Write an equation that relates known and unknown variables • Differentiate with respect to t • Plug in #’s and solve

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