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Suppose the joint cost function of a firm producing two products X and Y is given by C = 250X2 + 120Y2. Assuming that output of the two products is restricted at 4144. Using Lagrangian multiplier technique find the amounts of X and Y that will minimize cost and compute this cost. Also examine the cost implications of changing this optimal combination so as to produce equal amounts of both products.
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- Use the functions given below to solve the problem. C(x) represents the cost, in dollars, of x units of a product and R(x) represents the revenue, in dollars, from the sale of x units. Find the number of units that must be produced and sold in order to break even, C(x) = 72,000 + 43x and R(x) = 47x The number of units that must be produced and sold in order to break even is units.It is known that a certain company sells each kg of the product it manufactures at $80, it is also known that the total manufacturing cost "CT" is given by the function CT=(1/1000)x2 +100 , where "x" are the kg of product produced. a) How many units of "x" must the company sell to break even? Value = 100 points.b) How many units of "x" is the optimal quantity that should be sold to optimize the producer's profit? Value = 100 points.c) How many monetary units does that optimal profit for producers amount to? (remember, you will have to prove it mathematically either by the method of the first or by the method of the second derivative). Value = 100 points.The output (Q) of a production process is a function of two inputs (L and K) and K is given by the following relationship: Q = 0.50LK− 0.10L2− 0.05K2 The per-unit prices of inputs L and K are $20 and $25, respectively. The firm is interested in maximizing output subject to a cost constraint of $500. Use the lagrangian optimization techniques to find the following: How many units of L and K should be used by the firm? What is the total output of this combination? What is the marginal rate of substitution between L and K?
- The total cost function of a firm that produces its product on two assembly lines is given as TC (x,y) = 3x2+6y2-xy The problem facing the firm is to detrmine the least-cost combination of output on assembly lines x and y subject to the side condition that total output equal 20 units. Find the out put of x and y subject to a total of 20 units that minimize total cost by using both substitution and Lagrangian multiplier methodsMarket research for a certain ice cream company indicates that if the price per cup of ice cream is P50.00, the demand will be 1200 cups, whereas if the price is increased to P60.00, the demand will be 1100 cups. Therefore, at a production level of 200 cups of ice cream, the revenue generated by the company would be $130 for each additional cup of ice cream produced. Now, suppose it is known that the total cost of production of x cups of ice cream is given by C(x) = 3x + 25,500. (a) Find the value(s) of x at the break-even points. (b) Use marginal analysis to approximate the profit from the sale of the 500th cup of ice cream. Compare this value with the exact profit from the sale of the 500th cup.Pepper farming is carried out in a greenhouse. Proceeds from the sale of pepper, R,dollars per square meter is determined by the following function. R = 5T (1-e^-x) where T is the temperature set in the greenhouse (Celsius, C^0)and the amount of fertilizer per square meter (kilogram, kg) the costs are as follows: fertilizer cost per square meter is 20?, heating cost is 0.10^2.According to this information a) type the profit function of the manufacturer,?(?, ?).B) determine the end points of the profit function.c) show which endpoints you specify or which ones give the highest amount of fertilizer with the temperature value that makes the profit.
- Uver is a ride-sharing service that has become very popular in large cities. In one of these large cities, Uver is trying a new model: selling a bundle of 10 rides per week (instead of charging for each ride). Let y stand for the number of bundles, and suppose the weekly cost of providing a bundle of 10 rides is given by the following function: c(y) -10y²+200y+ 500. Assume all Uver drivers have the same cost function. 4th attempt See Hint The market for Uver rides in this city is competitive. Suppose there are 158 Uver drivers in the city. If the price of a bundle of rides is $164, the number of bundles of rides that will be provided is bundles.A small company produces organic cookies. When the price is $6 per dozen, the average daily sales has been 72 dozen cookies. When the price was decreased to $3 per dozen, the average daily sales increased to 110 dozen cookies. Assume that daily cookie sales is linearly related to price per dozen. Each dozen cookies has a variable cost of 87 cents to make, plus additional daily fixed costs of $84. Find a function that models the daily profit in terms of the sales x in dozens of cookies.The following production function has been estimated for wheat: W = 4 F0.5 L0.25 where, W = Wheat yield (kg/ha), F = Fertiliser (kg/ha) and L = Labour (hours/ha) a) Derive the expression for the least-cost expansion path given that fertilizers cost R12 per kg and labour cost R6 per hour b) If the wheat producer has a budget of R9 000 to spend on fertiliser and labour, how much of each input should he use and what is the quantity of wheat he will produce? c) Derive expressions for the value of marginal product (VMP) of fertiliser and labour given that wheat sells at R72 per kg.
- A company has the Cobb-Douglas production function z = 400x0.6 y0.4 where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Suppose labor costs Php 2000 per unit, capital costs Php 4000 per unit, and the total cost of labor and capital is limited to Php 1,000,000. Use the method of Lagrange Multipliers to solve the following problems. 1) Find the number of units of labor and the number of units of capital that maximize production. 2) How much should be allocated to labor and capital to maximize production? 3) What is the maximum production?A large company in the communication and publishing industry has quantified the relationship between the price of one of its products and the demand for this product as Price = 150 − 0.01 × Demand for an annual printing of this particular product. The fixed costs per year (i.e., per printing) = $50,000 and the variable cost per unit = $40. What is the maximum profit that can be achieved? What is the unit price at this point of optimal demand? Demand is not expected to be more than 6,000 units per year.Let p=17-√√x and C(x) = 323+2x, where x is the number of garden hoses that can be sold at a price of Sp per unit and C(x) is the total cost (in dollars) of producing x garden hoses. (A) Express the revenue function in terms of x. (B) Graph the cost function and the revenue function in the same viewing window for 0≤x≤ 289. Use approximation techniques to find the break-even points. (A) R(x)= Inc