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5 Use a spreadsheet to numerically verify the result of Exercises 1-55.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue,
Exercises 23-26
Minimizing cost. Assume that the costs of the materials for making the cylindrical container described in Exercise 48
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Chapter 2 Solutions
Calculus and Its Applications (11th Edition)
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- A firm produces output y from inputs x1 and x2 whose prices are given as w1 = 1 and w2 = 1 respectively. The production function is y = x11/2 x21/2 a) Write the Lagrangian function b) Find the firm’s cost function If you could write out the steps to the first part about how you got the function, that would be great. It would help me understand it.arrow_forwardThe long run. A chair manufacturer hires its assembly-line labour for $18 an hour and calculates that the rental cost of its machinery is $6 per hour. Suppose that a chair can be produced using 4 hours of labour or machinery in any combination. The firm is currently using 1 hour of labour for every 3 hours of machine time. (Assume that labour is on the horizontal axis and capital is on the vertical axis). 3. Graphically illustrate your answer by drawing an isoquant, an isocost line for the current combination of labour and capital and an isocost line for the optimal combination of labour and capital. An isocost corresponding to the optimal combination of labour and capital is [a vertical line, a horizontal line, an upward sloping straight line, an upward sloping curve which is not a straight line, a downward sloping straight line, a downward sloping curve which is not a straight line, L-shaped] has slope [ ] at the optimal combination of inputs An isoquant…arrow_forwardA patient starts with a concentration of medicine in her bloodstream equal to 10 milligrams per liter. Each day, the patient decomposes 60% of the medication in her bloodstream. However, the doctor gives her enough medication daily to increase the concentration in her bloodstream by 3 milligram per liter. What is the updating function and initial condition describing this DTDS? Graph the updating function, labeling axes appropriately. Find the equation of the solution function for this DTDS explicitly.arrow_forward
- Assume that the demand curve is P = f(Q) = 60 − 0.2Q and the supply curveis P=g(Q) = 30 + 0.3Q. Find the equilibrium price and quantity and also compute consumer and producer surplus by applying integral.arrow_forwardTraffic Control The accompanying figure shows the flow of downtown traffic during the rush hours on a typical weekday. Each avenue can handle up to 1500 vehicles/hour without causing congestion, whereas the maximum capacity of each street is 1000 vehicles/hour. The flow of traffic is controlled by traffic lights at each of the six intersections. (A) Suggest two possible traffic-flow patterns that ensure no traffic congestion. (B) Suppose the traffic flow along 9th Street between 5th and 6th Avenues, x6, is restricted due to sewer construction. What is the minimum permissible traffic flow along this road that will not result in traffic congestion?arrow_forwardA 500-gallon tank contains a 0.5 lb/gal salt solution with a volume of 200 gal. Then the tank is fed with a saline solution with a concentration of 1 lb/gal at a flow rate of 3 gal/min. This saline solution is stirred in a tank and has a tank outflow rate of 2 gal/min. Determine the model for the exit product concentration as a function of time and determine the value of the exit product concentration (in lb/gal) at 3.8 minutes.arrow_forward
- 2) A production function for a business follows the model P(L, K) = 0.1 L0.4K0.6 over the course of one year, where P represents the total value of all products the company makes, L represents the value of the labor from employees, and K represents the value of all the company's assets (buildings, machines, etc.), where everything is measured in millions of dollars. ap a. Find and ƏL ap ак' , and explain what they represent in this context. b. Evaluate the partial derivatives from part a when (L, K) = (3,4). If this was your business and your values, if you wanted to expand would it be better to expand in workforce (increase L) or expand in assets (increase K)?arrow_forwardAnswer part (iv)arrow_forwardThe daily world production of oil can be modelled using (30) ³0 0 (30) -50 V=10+100 where Vis volume of oil in millions of barrels, and t is time in years since 1 January 1980. From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as V=4.5 x 1.063t Use the models to show that the country's use of oil and the world production of oil will be equal during the year 2029.arrow_forward
- A large tank contains 400 liters of fluid in which 100 grams of salt is dissolved. Water is pumped into the tank at the rate of 5 L/min and the well mixed solution in the tank is pumped out at the rate of 10 L/min. Find the design equations for i). the liquid volume V (t) at any time t in the tank ii). the number of grams of salt S(t) in the tank as a function of time. Over what range of time will your equations be valid and why? What would be the salt content in the tank after one hour of operation?arrow_forwardA firm operates with the production function Q = K2L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. What is the input demand curve?arrow_forwardIn a hybrid-engine vehicle, energy from battery (“b" in Amperes) and from 95 octane unleaded gasoline ("g" in liters) are used intermittently to maximize the distance to be travelled (mileage) between fill- ups and charge-ups. If mileage (M) in kilometers is found out to be: M= 90 b+ 100 g - 3 b' - 5 g? - 2 b g find the values of “b" and “g" that will maximize mileage M, find the maximum mileage and prove that this is a maximum.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage