Question: A materials scientist in Brazil is developing a bio-based plastic that degrades at a rate proportional to the square of the time in weeks, $ t^2 $. The degradation function is modeled as $ D(t) = kt^2 + 5t $, where $ k $ is a constant. If after 3 weeks, the degradation is measured at 48 grams, what is the value of $ k $?

["Title: Discovering the Key Constant in Bio-Based Plastic Degradation Through Mathematical Modeling", "Meta Description:\nA Brazilian materials scientist is pioneering sustainable alternatives to conventional plastics by developing a bio-based polymer whose degradation follows the function $ D(t) = kt^2 + 5t $. With data showing 48 grams of degradation after 3 weeks, we solve for the unknown constant $ k $.", "---", "### Introduction\nIn the race to reduce plastic pollution, innovative materials science plays a crucial role. Recently, a team of researchers in Brazil has made significant progress in developing bio-based plastics designed not just to decompose, but to follow a precisely engineered degradation model. Their function, $ D(t) = kt^2 + 5t $, describes the mass of plastic degraded over time $ t $ in weeks, where $ k $ is a key constant yet to be determined.", "Recent experimental results show that after 3 weeks, $ D(3) = 48 $ grams. This real-world measurement provides the perfect opportunity to solve for $ k $ using basic algebra and mathematical modeling.", "---", "### Understanding the Degradation Model", "The degradation function is given by:\n$$\nD(t) = kt^2 + 5t\n$$\nHere:\n- $ D(t) $ represents the amount of plastic degraded after $ t $ weeks,\n- $ k $ is the specific degradation constant we need to find,\n- The $ 5t $ term models linear background degradation (e.g., environmental exposure), while $ kt^2 $ represents the accelerated breakdown over time — a hallmark of targeted bio-based materials.", "---", "### Applying the Given Data", "We know that at $ t = 3 $ weeks, $ D(3) = 48 $ grams. Substituting $ t = 3 $ into the equation:\n$$\nD(3) = k(3)^2 + 5(3) = 48\n$$\nSimplify:\n$$\n9k + 15 = 48\n$$", "Now solve for $ k $:\n$$\n9k = 48 - 15\n\Rightarrow 9k = 33\n\Rightarrow k = \frac{33}{9} = \frac{11}{3}\n$$", "---", "### Final Result", "The constant $ k $ in the degradation model is:\n$$\nk = \frac{11}{3}\n$$", "This value ensures the bio-based plastic degrades predictably, with degradation accelerating quadratically over time — a major breakthrough for sustainable material design.", "---", "### Conclusion", "By combining scientific innovation with rigorous mathematical modeling, researchers in Brazil are setting new standards in eco-friendly materials. Understanding constants like $ k $ not only validates experimental results but guides the next generation of biodegradable solutions. Whether you're a researcher, student, or sustainability advocate, this fusion of science and data highlights how precise modeling drives real-world impact.", "---", "Keywords: bio-based plastic, degradation model, materials science Brazil, k value, biodegradable polymer, synthetic D( t ) = kt² + 5t, sustainable materials, degradation rate, Brazil research, environmental science", "---", "For more updates on green innovation and materials engineering, stay tuned."]