math

a. A rectangular pen is built with one side against a barn. 1200

m of fencing are used for the other three sides of the pen. What dimensions maximize the area of theâ€‹ pen?
b. A rancher plans to make four identical and adjacent rectangular pens against aâ€‹ barn, each with an area of 25

msquared
.
What are the dimensions of each pen that minimize the amount of fence that must beâ€‹ used?

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1. (a) as with all these, divide the fence so that the lengths (1) equal the widths(2). Thus the pen is 600x300

(b) same problem, but now there are 4 widths, so the same principal applies. To minimize the fence, each pen has dimensions x and 25/x, where 4x = 5(25/x). That is, x^2 = 125/4, or x = 5√5/2

each pen is 5√5/2 by 2√5, with area of 25.

More generally, the fencing needed is
f = 4x + 5(25/x)
df/dx = 4 - 125/x^2
for minimum fencing, df/dx = 0, so x^2 = 125/4 as above.

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