Problems Suppose you want to make a fenced-in region like the one below. a. What

Question

Problems Suppose you want to make a fenced-in region like the one below. a. What is the total length of the fence (including the fence on the inside) in terms of x and y? b. What is the total area of the fenced-in region in terms of x and y? c. If you have to make the area of the fenced-in region 24 square feet, how should you build the Answer: Answer: fence in order to minimize the amount of fencing used? Give correct mathematical reasoning to show that your choice of measurements for x and y are indeed the minimal amount of fencing under the restriction. Justify and explain your work. Statel the objective and constraint functions, simplifying the objective function, stating a domain for the objective function and justifying the global behavior on that domain using calculus. Do not just write down a list of equations and formulas! You must write a narrative and mix words and explanations with symbols and equations. Summarize your solution in a least one complete sentence. Answer:

Explanation / Answer

a) Total length of the fence = Length of two horizontal strips + Length of three vertical strips

=> 2x + 3y

b) Total area of the fenced region in terms of x and y

Area = x * y

c)

xy = 24

y = 24/x

Length = 2x + 3y = 2x + 3(24/x) = 2x + 72/x

Taking the derivative of the length function wrt x, we get

d(Length)/dx = 2 - 72/x^2

Hence the value of x will be equal to 6

The value of y will be equal to 4

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