1). Consider the production function Q = 10K% L. (a) What is the output when K = 100000 and L = 243? (b) Use the marginal analysis to estimate Q(99995, 243) and Q(100000, 245). (c) Use a calculator to compute these two values of Q to three decimal places and compare these values with your estimates in (b). (d) How big must AL be in order for the difference between Q(100000, 243+AL) and its linear approximation Q(100000, 245) + ( AL, to differ by more than two units?
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- (i) Find the rate of change of the function f(x) =x + 2/ 1 − 8x with respect to x when x = 1.(ii) The number of units Q of a particular commodity that will be produced with Kthousand dollars of capital expenditure is modeled by Q(K) = 500 K2/3Suppose that capital expenditure varies with time in such a way that t months from nowthere will be K(t) thousand dollars of capital expenditure, whereK(t) =2t4 + 3t + 149 / t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be producedat this time?(b) At what rate will production be changing with respect to time 5 months from now?Will production be increasing or decreasing at this time?The cost, in millions of dollars, of building a three-story high school in NewYork State was estimated to be: 2C x x x x 1.7 0.14 0.0001 20 400where x is the number of thousands of square feet. Suppose that you arecontemplating building a for-profit three-story high school and estimate thatyour total revenue will be $0.2 million dollars per thousand square feet.(a) What is minimum area for a three-story high school in New YorkState, that this model covers? (b) What is the variable cost of building a three-story high school in NewYork State? (c) What is the fixed cost of building a three-story high school in NewYork State? (d) Determine the profit function,P x( ), for building a three-story highschool in New York State? . (e) Determine,P(250) , for building a three-story high school in New YorkState, and interpret the result. (f) How many thousands of square feet should a three-story high schoolbuilding in New York State be in order to break even?For 9 units of a product, a manufacturer's cost is C(9) dollars and revenue is R(g) dollars, with C(500) = 7500, R(500) = 9800, MC(500) = 15 and MR(500) = 25 . (a) What is the profit or loss at 4 500 ? The is (b) If production is increased from 500 to 501 units, by approximately how much does profit change? The change in profit is
- Hi, I have been trying to compute the optimal values from this lagrangian. But I can't seem to find a way to get the values from the answer key. I attached both the question and answers. Could you please explain to me how to get these values from the first-order conditions? Thank you very muchQ4 The output of a production process, Q, is given by the function KL Q = K+ 2L where K and L denote capital and labor. (a) Calculate the output when the capital and labor are 40 and 30 units, respectively. (b) Find the level of labor needed to produce 4 units of output when 10 units of capital are used. (c) Transpose this formula to express K in terms of Q and L. Q5 3 The equations defining a simple model of the economy are: C = 0.75Y + 18 I = 22 where C, Y and I denote consumption, national income and planned investment, respectively. Use this model to complete the following table of values and hence draw an accurate graph of C + I plotted against Y on the interval 0 < Y< 200. 100 200 C + I By drawing a second line on the diagram, calculate the equilibrium level of national income. Use your graph to explain what happens to the equilibrium level of national income when the marginal propensity to consume decreases with autonomous consumption fixed.A heavy equipment manufacturing company has produced its first unit using 3500 hours and $300,000 overhead cost. The next unit, i.e. the second unit took 3300 hours. Overhead cost for this unit was $200,000. Suppose the labor was $20 per hour. Being the production manager, you need to decide what would be the total production costs for additional 10 units after the completion of the second unit. Round the final amount to nearest integer. $1,597,832 C$1,003,183 (DNone of the above
- Q4 A business manager determines that t months after production begins on a new product, the number of units produced will be P thousand,where P(t) =6?2 + 5? /(? + 1)2 production in the long run (i) A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture.Turbulence makes it difficult to directly measure the thickness of the slick at the source (where x = 0),but for x > 0,it is found that y =0.5(x2 + 3x)/ x3 + x2 + 4x Required:a)Assuming the oil slick is continuously distributed,how thick would you expect it to be at the source?A firm has the following production function: Q = 9L2/3K1/3 Where Q is output, L is labor and K is Capital. Where Q is output, L is labor and K is Capital. (a) What sort of returns to scale characterize this production function? (b) Derive the marginal product of labor and capital. (c) What do we mean when we say the production function is linearly homogenous?A company finds that it can sell out a certain product that it produces, at the rate of Tk. 2 per unit. 1 It estimates the cost function of the product to be Tk. 1000 + 2 50 for q units produced. (i) Find the expression for the total profit, if q units are produced and sold. (ii) Find the number of units produced that will maximize profit. (iii)What is the amount of this maximum profit? (iv) What would be the profit if 6000 units were produced?
- Cost, revenue, and profit are in dollars and x is the number of units. A firm knows that its marginal cost for a product is МС- 4x+ 25, that its marginal revenue is MR%3D 75 - 6х, and that the cost of production of 60 units is $8,880. (a) Find the optimal level of production. units (b) Find the profit function. P(x) = (c) Find the profit or loss at the optimal level.The total cost (in hundreds of dollars) of producing x calculators per day is given by the equation. 20- 15- 10- C(x) = 6 + /2x + 32 Osx< 50 Perform the following calculations and interpret the results. 10 20 30 40 50 Production C'(x) =U %D Cost (hundred dollars)Ben's swim coach estimates that his time, t, in the 100-yard butterfly at a big upcoming meet as a function of the number of yards he swims per week, y, is t = 50+8e-0.00005y Based on this production function, his coach also estimates that if Ben swims 20,000 yards per week, his time will be 52.94 seconds, and he will finish tenth place. If he swims 40,000 yards per week, his time will be 51.08 seconds, and he will come in third. If he swims 60,000 yards per week, his time will be 50.40 seconds, and he will win the meet. Use calculus to determine Ben's marginal productivity of time given the number of yards he practices. Is there diminishing marginal productivity of practice yards? Ben's marginal productivity of swim time given y practice yards (MP) is (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a superscript can be created with the ^ MPy character.)