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- (b) A firm manufactures benches and tables: the quantity of benches manufactured is B and the quantity of tables manufactured is T. The total cost of manufacturing benches and tables is given by the equation: Cost = 3B2 + 5T²2 – 2BT. There is a shortage of lumber available to manufacture these items, and so, only 60 items in total have been promised to a customer. (i) Create the Lagrangian expression for Cost. (ii) The Lagrangian equations are given: 6B – 2T – 1 = 0 10T – 2B – A = 0 60 — В - Т%3D0 Calculate the minimum cost possible. (iii) If more lumber now becomes available, what selling price would be required to make a profit on an additional item produced?It is estimated that a grocer’s daily profit from the sale of two brands of cherry juice is given by thefunction:P(x,y) = (x - 30)(70 - 5x + 4y) + (y – 40)(80 + 6x – 7y) centswhere x is the price per can of Brand V Juice and y is the price per can of Brand Triple V Juice. CurrentlyBrand V Juice sells for 50 cents per can and Brand Triple V Juice sells for 52 cents per can.i. Use marginal analysis to estimate the change in the daily profit that will result if the grocer raises theprice of Brand Triple V by one cent per can while keeping the price of Brand V unchanged.ii. Find a stationary point of the daily profit function. Classify this stationary point.Find f'(x). T(x) = 2x° (x* - 2) 5 f'(x) =O
- Population of Fish in a Lake A lake is stocked with 100 fish. Let f(1) be the number of fish after i months, and suppose that y = f(1) satisfies the differential equation y' = .0004y(1000 – y). Figure 7 shows the graph of the solu- tion to this differential equation. The graph is asymptotic to the line y = 1000, the maximum number of fish that the lake can support. How fast is the fish population growing when it reaches one-half of its maximum population? 1000 800 600 400 200 10 15 Figure 7 Growth of a fish population.Given the functional relationship: y = 14x5/x2 find the value of y when x = 1.5After finding the optimal quantities of X1, X2 that maximize the function Z = (X1^0.2)*(X2^(1 -0.2)), subject to the contstraint 3X1 + 4X2 = 83, which is the X2* ? Give your answer to one decimal place (if you find 0.132: your answer should be 0.1).
- A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $5, and the society sells an average of 24 per week at a price of $9 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 3 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (Let x represent the price per feeder and P represent the profit.) P(x) = (b) What price should the society charge for each feeder to maximize profits? $ What is the maximum weekly profit? $A lake has two species of fish: bass and catfish. The number of bass and catfish living in the lake, B and C respectively, depend on the temperature of the lake, T (in degrees Farenheit). The functions modeling this dependency are given below: B(T) = 8T2 + 976T – 28890 С(Т) 3D — 2T?+244T — 6893 What are the two temperatures at which the number of bass in the lake is the same as the number of catfish? Enter exact expressions or round your answers to the nearest thousandth. T : TFind the values of x1 and x2 where the following two constraints intersect. (Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.) (1) 12x1 + 6x2 ≥ 64 (2) 3x1 + 6x2 ≥ 18
- Determine the following integral, using substitution method .Check the answer on your own (18 x (x+ 3).Use bordered determinants to check the following functions for quasiconcavity and quasiconvexity: a) Z=−x2−y2(x,y>0) b) Z=−(x+1)2−(y+2)2(x,y>0).Maximize the following equation: w (x, y) = -3x² + 30x – 6y² + 48y Subject to x + y = 5 O x=2.33; y=2.67 O x=3.33; y=2.67 O x=2.33; y=3.67 O x=12.33; y=12.67