Optimization: A farmer wants to fence an area of 1.5 million square feet in a rectangular field?

Question

and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? (Give the dimensions in increasing order.)

Everardo Feil · Accepted Answer

For the sake of argument, let's say that "length" is the dimension that the middle partition is parallel to, and the other two sides are the "width". Let L be the length and W be the width. Then the area is:LW = 1.5 x 10^6 square feet The total amount of fencing used, in feet, is going to be the sum of the four sides, plus the partition that runs down the middle:(L+L+W+W) + L = 3L + 2WUsing the first equation to substitute, you get3L + [2(1.5 x 10^6)/L] Now that you have the fencing equation in terms of just L, you can take the derivative, set it equal to 0, then solve for L. This tells you the value of L that minimizes the amount of fencing used, which in turn means that it minimizes the cost. To find W, that's just W = (1.5 x 10^6 square feet)/L.

Everardo Feil · Answer

Let the dimensions of the rectangle be x and y, so the length of fencing required is given by:L = 3x + 2yAnd the area, A is given byA = xy = 1500000 --> y = 1500000/xso L = 3x + 3000000/xdL/dx = 3 - 3000000/x^2 = 0 (at maximus)x^2 = 1000000x = 1000 --> y = 1500So your answer is 1500 ft, 1000 ft

Everardo Feil · Answer

p = width of the fieldq = lengthpq = 1.5 x 10^6 ft²Perimeter to be minimized = 2p + 3qP = 2p + 3(1.5 x 10^6 / p)P = 2p + (4.5 x 10^6)p^-1dP/dp = 2 + (4.5 x 10^6)(-1)p^-2dP/dp = 2 - (4.5 x 10^6) / p²Set dP/dp = 0 to find the minimum...2 - 4.5 x 10^6 / p² = 02p² - 4.5 x 10^6 = 02p² = 4.5 x 10^6p² = 9 x 10^6p = √(9 x 10^6)p = 3000 ft

Everardo Feil · Answer

enable container have sides L and B and enable fence be parallel to section L Perimeter + divider = 3L + 2B....................(a million) element of container = L*B = 3000000 So, B = 3000000/L replace this value of B in (a million) Fencing = 3L + (6000000/L) or (3L^2 + 6000000) / L

Mr. Durward Cummerata

Optimization: A farmer wants to fence an area of 1.5 million square feet in a rectangular field?

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