We are given a water treatment plant that processes 10,000 cubic meters of contaminated water per day. The energy consumption (in kilowatt-hours) of the plant is given by the function E(t) = 0.005t^2 + 20t + 500, where t represents the time in hours since the plant started operating.
a. To minimize the energy consumption, we need to find the time t at which the function E(t) attains its minimum value.
b. To optimize the efficiency of the water treatment process, we need to determine the optimal flow rate of water through different stages of the treatment plant. The flow rate (in cubic meters per hour) at a particular stage is given by the function f(t) = 1000 + 50t, where t represents the time in hours since the water entered that stage. Determine the time at which the flow rate reaches its maximum and analyze its implications for the plant's overall performance.
c. Contaminant Removal Analysis One of the critical objectives of the water treatment plant is to remove contaminants from the water effectively. The concentration of a particular contaminant (in milligrams per liter) in the treated water is given by the function C(t) = 200e^(-0.1t), where t represents the time in hours since the start of the treatment process. Determine the rate at which the concentration of the contaminant is decreasing after 2 hours and analyze the implications for the plant's performance.
d. Efficient distribution of treated water is crucial for the water treatment plant's functionality. Suppose the water distribution system is represented by a vector field V(x, y) = (2x, y^2), where x and y represent the coordinates in a two-dimensional space. Determine the path along which the water flows most rapidly and analyze the implications for the plant's water distribution design.