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- A piece of wire of length 58 is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square?Solve min x1-x2-2x3 problem by means of Lagrange multipliers under the constraints of x1+x2+x3=5 and x12+ x22 =4. Explain what the values of Lagrange multipliers mean?A piece of wire of length 60 is cut, and the resulting two pieces are formed to make a circle and square. What should the wire be cut to (a) maximize and (b) minimize the combined area of the circle and the square?
- 7. Use Lagrange multipliers to give an alternate solution. Find two positive numbers whose product is 100 and whose sum is a minimum.Which delays should he specially handle to minimize?a) The base chassis of a manufactured product is made up of numerous components. There can sometimes be differences in dimensions between components, requiring the partially assembled product to be sent to another department for extra fitting work because some components cannot be fitted in place. The majority of fitting problems are caused by two particular components (identified as B and F): the chances of a randomly chosen component of type B not fitting are 3.7%, and the chances of a randomly chosen component of type F not fitting are 2.8%. (i) Given that the problems with fitting these two components are independent of each other and that components are selected randomly for assembly on each base chassis, what is the probability that both components B and F will have problems fitting on a given chassis? ii) How likely is it that there will be problems fitting components B and F to a given chassis under the same assumptions? (b) (i) The probability distribution of the number of…
- a) The base chassis of a manufactured product is made up of numerous components. There can sometimes be differences in dimensions between components, requiring the partially assembled product to be sent to another department for extra fitting work because some components cannot be fitted in place. The majority of fitting problems are caused by two particular components (identified as B and F): the chances of a randomly chosen component of type B not fitting are 3.7%, and the chances of a randomly chosen component of type F not fitting are 2.8%. (i) Given that the problems with fitting these two components are independent of each other and that components are selected randomly for assembly on each base chassis, what is the probability that both components B and F will have problems fitting on a given chassis? ii) How likely is it that there will be problems fitting components B and F to a given chassis under the same assumptions? (b) (i) The probability distribution of the number of…The demand for a commodity is given by Q = β0 + β1P + u, whereQ denotes quantity, P denotes price, and u denotes factors other thanprice that determine demand. Supply for the commodity is given byQ = ϒ0 + ϒ1P + v, where v denotes factors other than price that determinesupply. Suppose that u and v both have a mean of zero, have variancesσ2u and σ2v, and are mutually uncorrelated.a. Solve the two simultaneous equations to show how Q and P dependon u and v.b. Derive the means of P and Q.c. Derive the variance of P, the variance of Q, and the covariancebetween Q and P.d. A random sample of observations of (Qi, Pi) is collected, and Qi isregressed on Pi. (That is, Qi is the regressand, and Pi is the regressor.)Suppose that the sample is very large. i. Use your answers to (b) and (c) to derive values of the regressioncoefficients. ii. A researcher uses the slope of this regression as an estimate of theslope of the demand function (β1). Is the estimated slope too largeor too small?In a study of the causes of bearing wear, a machine was run 24 times, with various loads (denoted x1), oil viscosities (x2), and ambient temperatures (x3). The wear, denoted y, was modeled as y = β0 + β1x1 + β2x2 + β3x3 + ε. When this model was fit to the data, the sum of squares for error was SSE = 9.37. Then the reduced model y = β0 + β1x1 + β2x2 + β3x3 was fit, and the sum of squares for error was SSE = 27.49. Is it reasonable to use the reduced model, rather than the model containing all the interactions, to predict wear? Explain.
- Suppose there are four cities at locations x1 = 2; x2 = 5; x3 = 9, and x4 = 11, requiring respectively Ni = 6; 2; 3; 7 visits, respectively. Suppose that the cost-per-mile to ship goods is $4 between x1 and x2, but $8 between x2 and x3, and $10 between x3 and x4. What is the cost of keeping the PDC at x = 6? What about at x = 10?Suppose the feasible region has four corners at these points (0,0), (8,0), (0,12), and (4,8). If the profit formula is P = $2x + $4y, what is the maximum profit possible?What are the pros and cons of using Factorial Designs? What can and should you do to minimize the cons? (if you can also answer, but first Q is most important: What are post hoc tests and when should they be used? Why are they important?)