KW; = power generated by turbinė 2 in kilowatts QT = total flow through the station in cubic feet per second 1. If all three turbines are being used, we wish to determine the flow Q, to each turbine that will give the maximum total energy production. Our limitations are that the flows must sum to the total incoming flow and the given domain restrictions must be observed. Consequently, use Lagrange multipliers to find the values for the individual flows (as functions of Qr) that maximize the total energy production KW₁+KW₂+KW3 subject to the constraints Q1 + Q2 +Qs = QT and the domain restrictions on each Q.. 2. For which values of QT is your result valid? 3. For an incoming flow of 2500 ft³/s, determine the distribution to the turbines and verify (by trying some nearby distributions) that your result is indeed a maximum. 4. Until now we have assumed that all three turbines are operating; is it possible in some situations that more power could be produced by using only one turbine? Make a graph of the three power functions and use it to help decide if an incoming flow of 1000 ft³/s should be distributed to all three turbines or routed to just one. (If you determine that only one turbine should be used, which one would it be?) What if the flow is only 600 ft³/s? 5. Perhaps for some flow levels it would be advantageous to use two turbines. If the incoming flow is 1500 ft³/s, which two turbines would you recommend using? Use Lagrange multipliers to determine how the flow should be distributed between the two turbines to maximize the energy produced. For this flow, is using two turbines more efficient than using all three? 6. If the incoming flow is 3400 ft³/s, what distribution would you recommend to the station management? Rama/Shutterstock.com Romaset/Shutterstock.com The power station has three different hydroelectric turbines, each with a known (and unique) power function that gives the amount of electric power generated as a function of the water flow arriving at the turbine. The incoming water can be apportioned in different volumes to each turbine, so the goal of this project is to determine how to distribute water among the turbines to give the maximum total energy production for any rate of flow. Using experimental evidence and Bernoulli's equation, the following quadratic models were determined for the power output of each turbine, along with the allowable flows of operation: 6 KW₁ = (-18.89 +0.1277Q1-4.08.10 Q) (170 - 1.6 · 10¯*Q) KW2 = (-24.51 +0.1358Q2-4.69-10 Q¹²) (170 — 1.6 · 10¯*Q) KW3 = (-27.02 +0.1380Q3 -3.84-10-5Q) (170 - 1.6-10-ºQ) where 250 Q1 <1110, 250 Q2 <1110, 250 <3 < 1225 Qi = flow through turbine i in cubic feet per second KW = power generated by turbine i in kilowatts

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

Suppose one canoe rents for 40,and2 is taken off the price for each additional canoe rented by a ground. What size group gives the most income? Assume that there are 20 canoes available.