# Question #b9993

##### 1 Answer

The total length of the partition should be

#### Explanation:

Without loss of generality let us assume that the pen is divided as shown:

Let us set up the following variables:

# {(x, "Total height of the partition (feet)"), (y, "Total length of the partition (feet)"), (A, "Total Area enclosed by the partition (sq feet)") :} #

Our aim is to find

Now, the total perimeter is given as

# 5x + 2y=500 #

# :. 2y=500 - 5x#

# :. y=250 - 5/2x# ..... [1]

And the total Area enclosed by the pen is given by:

# A =xy #

And substitution of the first result [1] gives us:

# A =x(250 - 5/2x) #

# \ \ \ = 250x - 5/2x^2 #

We no have the Area, A, as a function of a single variable, so Differentiating wrt

# (dA)/dx = 250 -5x # ..... [2]

At a critical point we have

# 250-5x = 0 #

# :. \ \ \ \ \ 5x = 250 #

# :. \ \ \ \ \ \ \ x = 50 #

And substituting

# y=250 - 5/2(50)#

# \ \ =250 - 125#

# \ \ =125#

We should check that

# (d^2A)/dx^2 = -5 < 0 # when#x=50#

Confirming that we have a maximum area, given by:

# A = (50)(125) = 6250 " feet"^2 #

We can visually verify that this corresponds to a maximum by looking at the graph of

graph{250x - 5/2x^2 [-100, 200, -100, 7000]}