1. **Problem Statement:** We analyze the population growth of Green Leaf Beetles and Red Bark Beetles over time $t$ (in hours). 2. **Given Functions:** - Green Leaf Beetles: $g(t) = t^2 + 30t + 200$ - Red Bark Beetles: $r(t) = t^2 + 180t + 150$ --- ### 1.1 Initial Population - Initial population means population at $t=0$. - For Green Leaf Beetles: $g(0) = 0^2 + 30\times0 + 200 = 200$ - For Red Bark Beetles: $r(0) = 0^2 + 180\times0 + 150 = 150$ ### 1.2 Time when populations are equal - Set $g(t) = r(t)$: $$t^2 + 30t + 200 = t^2 + 180t + 150$$ - Simplify: $$30t + 200 = 180t + 150$$ $$200 - 150 = 180t - 30t$$ $$50 = 150t$$ $$t = \frac{50}{150} = \frac{1}{3} \approx 0.333 \text{ hours}$$ ### 1.3 Graphical Representation - The populations are equal at $t \approx 0.333$ hours. - On a graph, the curves $g(t)$ and $r(t)$ intersect at this point. ### 1.4 Behavior of Green Leaf Beetles as $t \to \infty$ - The dominant term is $t^2$. - As $t$ increases, $g(t) \approx t^2$ grows without bound (quadratic growth). - So, population increases rapidly to infinity. ### 1.5 Type of function for Red Bark Beetles - $r(t) = t^2 + 180t + 150$ is a quadratic function (polynomial of degree 2). ### 1.6 Domain and Range of $r(t)$ - Domain: All real numbers $t$ since polynomial functions are defined everywhere. - Range: Since the leading coefficient of $t^2$ is positive, the parabola opens upward. - Vertex at $t = -\frac{b}{2a} = -\frac{180}{2} = -90$. - Minimum value at vertex: $$r(-90) = (-90)^2 + 180(-90) + 150 = 8100 - 16200 + 150 = -5950$$ - Range is $[-5950, \infty)$. ### 1.7 New growth function for Red Bark Beetles - New function: $g_{new}(t) = 150 e^{2t}$ - Find $t$ when $g_{new}(t) > r(t)$: $$150 e^{2t} > t^2 + 180t + 150$$ - At $t=0$, $g_{new}(0) = 150$, $r(0) = 150$ equal. - For large $t$, exponential dominates polynomial. - Approximate by testing values or solving numerically: - At $t=1$: $150 e^{2} \approx 150 \times 7.389 = 1108.35$, $r(1) = 1 + 180 + 150 = 331$ so $g_{new}(1) > r(1)$. - So, $g_{new}(t)$ surpasses $r(t)$ shortly after $t=0$. --- **Final answers:** - Initial populations: Green Leaf Beetles = 200, Red Bark Beetles = 150 - Populations equal at $t = \frac{1}{3}$ hours - Graph intersection at $t \approx 0.333$ - Green Leaf Beetles grow quadratically to infinity as $t \to \infty$ - Red Bark Beetles growth modeled by quadratic function - Domain of $r(t)$: $(-\infty, \infty)$; Range: $[-5950, \infty)$ - New growth function surpasses initial function just after $t=0$