Question

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?

Answer 1

Answer:

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