1. **Problem Statement:** Water is flowing out of tanks P, Q, and R. We need to analyze the volume of water over time based on the graph. 2. **Given Information:** - Volume of water in each tank at time $x=0$ minutes is 8000 cm³. - At $x=40$ minutes, volumes are approximately: - Tank P: 4000 cm³ - Tank Q: 2000 cm³ - Tank R: 0 cm³ 3. **a. Initial volume in each tank:** At $x=0$, all tanks have 8000 cm³. 4. **b. Volume at 40 minutes:** - Tank P: 4000 cm³ - Tank Q: 2000 cm³ - Tank R: 0 cm³ 5. **c. Linear pattern identification:** Each tank's volume decreases linearly over time, so all are shrinking patterns. This means water is flowing out steadily from each tank. 6. **d. Write equations in the form $y=mx+b$:** - $b$ is the initial volume at $x=0$, so $b=8000$ for all. - $m$ is the rate of change (slope), calculated as $m=\frac{\Delta y}{\Delta x}$. For Tank P: $$m_P=\frac{4000-8000}{40-0}=\frac{-4000}{40}=-100$$ Equation: $$y_P=-100x+8000$$ For Tank Q: $$m_Q=\frac{2000-8000}{40-0}=\frac{-6000}{40}=-150$$ Equation: $$y_Q=-150x+8000$$ For Tank R: $$m_R=\frac{0-8000}{40-0}=\frac{-8000}{40}=-200$$ Equation: $$y_R=-200x+8000$$ 7. **e. Fastest rate of water flowing out:** Tank R with rate $-200$ cm³/min (largest magnitude). 8. **f. Slowest rate of water flowing out:** Tank P with rate $-100$ cm³/min (smallest magnitude). **Final answers:** - a. Initial volumes: 8000 cm³ each. - b. Volumes at 40 min: P=4000, Q=2000, R=0 cm³. - c. All shrinking linear patterns. - d. Equations: - $y_P=-100x+8000$ - $y_Q=-150x+8000$ - $y_R=-200x+8000$ - e. Fastest outflow: Tank R at 200 cm³/min. - f. Slowest outflow: Tank P at 100 cm³/min.