1. **Problem:** Simplify and analyze the quadratic function $C = 8n^2 - 176n + 1800$. 2. **Problem:** Analyze the cubic function $f(x) = x^3 - 15x$. 3. **Problem:** Calculate the definite integral $\int_0^{20} (60 - 0.5t) \, dt$. 4. **Problem:** Analyze the cubic function $v(t) = t^3 - 0.5t^2 + 4$ representing data flow rate. --- ### 1. Quadratic function $C = 8n^2 - 176n + 1800$ 1. The formula is a quadratic polynomial in $n$. 2. To analyze, we can factor or find vertex. 3. Factor out common factor 4: $$C = 4(2n^2 - 44n + 450)$$ 4. Use quadratic formula for roots: $$n = \frac{44 \pm \sqrt{(-44)^2 - 4 \cdot 2 \cdot 450}}{2 \cdot 2} = \frac{44 \pm \sqrt{1936}}{4}$$ 5. Calculate discriminant: $$1936 = 44^2 - 3600 = 1936$$ 6. Roots: $$n = \frac{44 \pm 44}{4}$$ 7. So, $$n_1 = \frac{44 + 44}{4} = 22, \quad n_2 = \frac{44 - 44}{4} = 0$$ ### 2. Cubic function $f(x) = x^3 - 15x$ 1. To find critical points, compute derivative: $$f'(x) = 3x^2 - 15$$ 2. Set derivative to zero: $$3x^2 - 15 = 0 \Rightarrow x^2 = 5 \Rightarrow x = \pm \sqrt{5}$$ 3. Evaluate $f(x)$ at critical points: $$f(\sqrt{5}) = (\sqrt{5})^3 - 15\sqrt{5} = 5\sqrt{5} - 15\sqrt{5} = -10\sqrt{5}$$ $$f(-\sqrt{5}) = -5\sqrt{5} + 15\sqrt{5} = 10\sqrt{5}$$ ### 3. Definite integral $\int_0^{20} (60 - 0.5t) \, dt$ 1. Integrate term by term: $$\int (60 - 0.5t) dt = 60t - 0.5 \frac{t^2}{2} + C = 60t - 0.25 t^2 + C$$ 2. Evaluate from 0 to 20: $$\left[60t - 0.25 t^2\right]_0^{20} = (60 \cdot 20 - 0.25 \cdot 400) - (0) = 1200 - 100 = 1100$$ ### 4. Cubic function $v(t) = t^3 - 0.5t^2 + 4$ 1. This function models data flow rate in kilobytes per second. 2. Derivative to find rate of change: $$v'(t) = 3t^2 - 1t = 3t^2 - t$$ 3. Set derivative to zero for critical points: $$3t^2 - t = 0 \Rightarrow t(3t - 1) = 0 \Rightarrow t = 0 \text{ or } t = \frac{1}{3}$$ 4. Evaluate $v(t)$ at critical points: $$v(0) = 0 - 0 + 4 = 4$$ $$v\left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^3 - 0.5 \left(\frac{1}{3}\right)^2 + 4 = \frac{1}{27} - 0.5 \cdot \frac{1}{9} + 4 = \frac{1}{27} - \frac{1}{18} + 4 = \frac{2}{54} - \frac{3}{54} + 4 = -\frac{1}{54} + 4 \approx 3.9815$$ --- **Final answers:** 1. Roots of $C$: $n=0$ and $n=22$. 2. Critical points of $f(x)$ at $x=\pm \sqrt{5}$ with values $f(\sqrt{5}) = -10\sqrt{5}$ and $f(-\sqrt{5}) = 10\sqrt{5}$. 3. Definite integral value: $1100$. 4. Critical points of $v(t)$ at $t=0$ and $t=\frac{1}{3}$ with values approximately $4$ and $3.9815$ respectively.