I can't understand the textbook answer of the length equal to 190 m approximately and the width equal to 63 m in the following problem.
An animal breeder wishes to create five adjacent rectangular pens, each with an area of 2400 m^2. To ensure that the pens are large enough for grazing, the minimum for either dimension must be 10 m. Find the dimensions required for the pens to keep the amount of fencing used to a minimum.
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The breeder has to construct 5 rectangular pens which are adjacent to each other. Let the sides of the pens have lengths S1 and S2 and let the common side S2.
The length of fence required to construct this is 10*S1 + 6*S2
The area of each pen is 2400 m^2
=> S1*S2 = 2400
=> S1 = 2400/S2
The length of fence can be rewritten as : 24000/S2 + 6*S2
The length has to be minimized. This is done by finding the derivative and solving it for s
-24000/S2^22 + 6 = 0
=> S2^2 = 24000/6 = 4000
=> S2 = sqrt 4000 = 63.52 m
S1 = 2400/S2 = 37.78
For each pen the length is 63.52 m and the width is 37.78 m. Multiplying 37.78 by 5 gives approximately 180 m. So the dimensions in your book are the width of each pen and the length made up by adding the shorter side of each pen.
To minimize the length of the fence required the length of each pen should be 63.52 m and the width of each pen should be 37.78 m.
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