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]}