(a) Suppose a Firm produce output using the production function Q-5KL. Wage rate for hiring labor 200 an hour and cost of using capital is 100 per hour. (1) Find out the optimal quantities of labor (L) & capital (K’) to produce 1000 unit of output?. (i) Also show the result graphically and interpret.
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- 6) use differentials to find approximate valueA window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window.A mass of 1 kgkg is attached to the end of a spring whose restoring force is 180 NmNm. The mass is in a medium that exerts a viscous resistance of 156 NN when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object. Suppose the spring is stretched 0.07 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7sin(3t)7sin(3t) NN at time tt seconds. Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.)
- Express the mathematical relationship between stress and strain in case of elastic material.The company is conducting set of experiments to maximize the time needed to discharge the battery. It was found that the time (t) is proportional to different powers of acceleration (a) , mass (m), and energy (E) of the battery. Use dimensional homogeneity to find the formula of the time and the value of overall dimensionless proportional constant using t=24 hours, the voltage is 338 V, the Energy is 71 kWh, velocity is= 1D km/hr, and the mass is 14D-pound.One method of slowing the growth of an insect populationwithout using pesticides is to introduce into the populationa number of sterile males that mate with fertile femalesbut produce no offspring. If P represents the number offemale insects in a population, S the number of sterile malesintroduced each generation, and r the population’s naturalgrowth rate, then the female population is related to time tby equation given at the end. Suppose an insect population with 10,000 females grows ata rate of r = 0.10 and 900 sterile males are added. Evaluatethe integral to give an equation relating the female populationto time. (Note that the resulting equation can’t besolved explicitly for P.)
- Suppose you are climbing a hill whose shape is given by the equationz= 1000−0.01x2−0.02y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (50, 80, 847). The positive x-axis points east and the positive y-axis points north. (a) If you walk northwest, will you start to ascend or descend? At what rate? (b) In what direction is the slope largest? Why?There is a tank in the shape of a paraboloid: Water is being pumped into the tank at a constant rate of 2π cubic inches per minute. If y denotes the water level, then the volume of water is given by V = πy2 / 2. (a) Suppose that the height of the paraboloid is 10 inches. Let a denote the rate of change of water level when y = 1 inch and let b denote the rate of change of water level when y = 5 inches. Without computing anything, do you suspect that a or b is greater? (b) Confirm your suspicion by explicitly computing a and bA tank initially contains 200 liters of brine with 20 kg of salt solution. Brine containing 0.25 kg per liter of salt flows into the tank at the rate of 3 liters/min, and the mixture kept uniform by stirring, flows out at the rate of 2 liters/minute. Determine the time when the concentration of salt becomes 0.15kg/liter.
- A mass of 1.75 kg is attached to the end of a spring whose restoring force is 100 N/m. The mass is in a medium that exerts a viscous resistance of 15 N when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object.Suppose the spring is stretched 0.03 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 3sin(2t) NN at time tt seconds.Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.) Enter exact value for the constants, or 8 decimal places. u(t) =Suppose that the demand for a product is given by q2p=6,000, where q represents the number of units of the product demanded and p represents the price per unit in dollars. a. Compute the elasticity of demand when the number of units demanded is 50 and the price per unit is $16. b. Is the demand elastic, inelastic, or unitary elastic? Could you look over my work? This should be solved using implicit differentiation. You solve for dq/dp and plug it into the function n=(-p/q)(dq/dp)A window has the shape of a square surmounted by semi-circle. The base of the window has a width of 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window.