10. Do only one of Optimization problems (You may do one more for extra credit)

Question

10. Do only one of Optimization problems (You may do one more for extra credit) Show all your work. (picture, names of the variables, relations of the variables, the function you want to optimize, derivative, critical numbers.... (1) A rectangular field with a partition is to be enclosed with a fence against an existing wall, so that no fence is needed on that side. The material for the fence cost $3/ft. for the two end sides and the partition perpendicular to the wall and S5/ft. for the side parallel to the wall. Find: (a) the dimensions of the field of largest area that can be enclosed for $360, (b) the maximum area (2) A rectangular field of 1350 ft2 is to be enclosed with a fence against an existing wall, so that no fence is needed on that side. The material for the fence cost $3/ft. for the two end sides perpendicular to the wall and S4/ft. for the side parallel to the wall. Find: (a) the dimensions of the field of which minimized the cost, (b) the minimum cost lo 3) F ind the point on the parabolax 0 that is closest to point P(0,-3).

Explanation / Answer

I am solving the first question (Q1 in this case) as per Chegg guidelines, post multiple question to get the remaining answers

Q1)

Let L = length of 3$ sides

W = length of $5 side

Cost: 2L($3) + W($5) = 360$

6L + 5W = 360

Area = Length * Width = L * (360-6L)/5

Taking the derivative of the area wrt L, we get

360 - 12L = 0, L = 30

6 * 30 + 5 * W = 360

5W = 180

W = 36

Hence the dimensions of the largest area will be 30 ft X 36 ft

Area = Length * Width = 30 ft * 36 ft = 1080 ft^2

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