a. A rectangular pen is built with one side against a barn. 1900 m of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 msquared. What are the dimensions of each pen that minimize the amount of fence that must be used?

Question

Physics

Bryant Porter

16 August, 07:27

a. A rectangular pen is built with one side against a barn. 1900 m of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 msquared. What are the dimensions of each pen that minimize the amount of fence that must be used?

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Answers (1)

Know the Answer?

Nickolas Tanner · Answer 1

475 m, 950 m

Explanation:

Let l be the length of the side perpendicular to the barn.

1900-2l = length of the side parallel to the barn

Area A = l (1900-2l)

A = 1900l-2l^2

now, the maximum value of l (the equation being quadratic)

l_max = - b/2a

a = 2

b=1900

l_max = - 1900/4 = 475 m

then 1900-2l = 1900-2 * (475) = 950 m

So, the dimensions that maximize area are

950 and 475

Now. A_max = - 2 (l_max) ^2+1900*l_max

A_max = - 2 (475) ^2+1900*475

A_max = 451250 m^2

or, 475*950 = 451250 m^2