5A plant biologist is studying a genetically modified crop that grows at a rate modeled by the function $ f(t) = 3t^2 + 2t + 1 $, where $ t $ is time in weeks. What is the total growth from week 1 to week 4?

["Title: Analyzing the Growth of a Genetically Modified Crop: A 5A Plant Biologist’s Study", "In the field of modern agriculture, understanding the growth patterns of genetically modified crops is essential for optimizing yields and ensuring food security. One such breakthrough is a unique crop whose growth rate is mathematically modeled by the quadratic function:", "$$\nf(t) = 3t^2 + 2t + 1\n$$", "where $ t $ represents time in weeks. This function provides the growth rate (in centimeters per week) of the crop. A dedicated 5A plant biologist has recently focused on this model to analyze the total growth over a four-week period—from week 1 to week 4.", "---", "### Understanding the Model", "The function $ f(t) = 3t^2 + 2t + 1 $ describes the instantaneous growth rate at any given week $ t $. To determine the total growth over time, we must calculate the definite integral of $ f(t) $ from $ t = 1 $ to $ t = 4 $, which gives the total accumulation of growth over that interval.", "---", "### Calculating Total Growth", "We compute:", "$$\n\ ext{Total Growth} = \int_{1}^{4} (3t^2 + 2t + 1) , dt\n$$", "Break this into smaller integrals:", "$$\n\int_{1}^{4} 3t^2 , dt + \int_{1}^{4} 2t , dt + \int_{1}^{4} 1 , dt\n$$", "Now evaluate each:", "1. $$\n\int_{1}^{4} 3t^2 , dt = 3 \left[ \frac{t^3}{3} \right]<em 1="1">1^4 = \left[ t^3 \right]_1^4 = 4^3 - 1^3 = 64 - 1 = 63\n$$", "2. $$\n\int \right]}^{4} 2t , dt = 2 \left[ \frac{t^2}{2<em 1="1">1^4 = \left[ t^2 \right]_1^4 = 4^2 - 1^2 = 16 - 1 = 15\n$$", "3. $$\n\int 1 , dt = [t]_1^4 = 4 - 1 = 3}^{4\n$$", "---", "### Summing the Results", "Add the three components:", "$$\n63 + 15 + 3 = 81\n$$", "---", "### Conclusion", "From week 1 to week 4, the genetically modified crop in this study accumulates a total growth of:", "$$\n\boxed{81 \ ext{ cm}}\n$$", "This analysis underscores the power of mathematical modeling in plant biology and highlights how precise measurements and integration can reveal critical insights into crop development. As genetically modified crops continue to evolve, such quantitative approaches remain vital for advancing sustainable agriculture.", "For 5A plant biologists and agricultural researchers, understanding growth functions like $ f(t) $ enables smarter decisions—from breeding programs to irrigation planning—ultimately contributing to global food security."]