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Solution Manual for Mathematical Modeling, 4th Edition

Solution Manual for Mathematical Modeling, 4th Edition is an essential tool for mastering your textbook, offering detailed solutions and helpful explanations.

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1. An automobile manufacturer makes a profit of $1,500 on the sale of a certain model.It is estimated that for every $100 of rebate, sales increase by 15%.(a) What amount of rebate will maximize profit? Use the five-step method, and model asa one-variable optimization problem.Step 1: Ask the question.Variables:r= rebate ($)s= number of cars soldP= profit ($)Assumptions:s= s_0 (1+0.15(r/100))P= (1500-r) ss>= 0 , 0<= r <=1500where the constant s_0 is the number of sales without any rebateObjective:Maximize P.Step 2: Select the modeling approach.We will model this problem as a one variable optimization problem. See text p. 6.Step 3: Formulate the model.Let x=r and y=P, and writey = f(x) = (1500-x) s_0 (1+0.15(x/100)).Our goal is to maximize f(x) over the interval [0, 1,500].Step 4: Solve the model.Compute f '(x) = 1500 s_0 (0.0015) + (-1) s_0 (1+0.15(x/100)) = 0 at x = 416.6667,f(x) = 1760.42 and since the graph of f(x) is a parabola we know this is the globalmaximum.Step 5: Answer the question.According to this model, the optimal policy is to offer a rebate of around $420, whichshould result in about a 17% increase in profits as compared to no rebate.(b) Compute the sensitivity of your answer to the 15% assumption. Consider both theamount of rebate and the resulting profit.Generalize the model from part (a) and lety = f(x) = (1500-x) s_0 (1+e x)

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Subject
Mathematics
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Mathematical Modeling by Mark M. Me...

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