Read and analyze each situation carefully and apply your learnings on representing real-life situations involving functions including piecewise.1. Contaminated water is subjected to a cleaning process. The concentration of the pollutants is initially 5 mg per liter of water. If the cleaning process can reduce the pollutant by 10% each hour, define a function that can represent the concentration of pollutants in the water in terms of the number of hours that the cleaning process has taken place.2. During typhoon Ambo, PAGASA tracks the amount of accumulating rainfall. For the first three hours of typhoon, the rain fell at a constant rate of 25mm per hour. The typhoon slows down for an hour and started again at a constant rate of 20 mm per hour for the next two hours. Write a piecewise function that models the amount of rainfall as function of time.
Answer (1)
Answer:Step-by-step explanation:1. Contaminated Water Cleaning ProcessTo determine the concentration of pollutants over time, we use an exponential decay model. The initial concentration is 5 mg/L, and it decreases by 10% each hour. This means that each hour, 90% of the pollutant remains.Let C(t) be the concentration of pollutants in mg/L after t hours.The function is:C(t)=5(0.90)t 2. Rainfall During Typhoon AmboThis situation requires a piecewise function because the rate of rainfall changes over different time intervals.Let R(t) be the total amount of rainfall in mm after t hours.For the first three hours (0 ≤ t ≤ 3): The rain fell at a constant rate of 25 mm per hour.R(t)=25tFor the next hour (3 < t ≤ 4): The typhoon slowed down, meaning no additional rain fell during this hour. The amount of rain accumulated up to t=3 hours is 25×3=75 mm. So, for this hour, the rainfall remains constant at 75 mm.R(t)=75For the next two hours (4 < t ≤ 6): The rain started again at a constant rate of 20 mm per hour. We need to add this new accumulation to the 75 mm already accumulated. The time elapsed in this phase is (t−4) hours.R(t)=75+20(t−4)Combining these, the piecewise function is:R(t)= ⎧25t if 0≤t≤3⎨75 if 3<t≤4⎩75+20(t−4) if 4<t≤6
