Question

Assume you have 12 yards of fencing material available What is the largest area, measured in square yards, of a rectangular enclosure that has a fenced center divider (so that the final enclosure looks like a domino from above)

Answer 1

Answer:

The maximum area is when the largest side of the rectangle is 3 yards and the shortest side of the rectangle is 2 yards.

Step-by-step explanation:

If the final enclosure will look froma above like a domino (rectangular with a center divider).

If we name a and b the sides of the rectangle, we have the area as:

$S=a\cdot b$

If we divide the center of the rectangle with a size b fence (one of the sides of the rectangle), the perimeter used is:

$P=2a+2b+b=2a+3b=12\\\\a=\frac{12-3b}{2}=6-1.5b$

To maximize the area, we derive and equal to zero (after replacement of a by the last equation):

$S=(6-1.5b)\cdot b=-1.5b^2+6b\\\\dS/db=-1.5*2*b+6=0\\\\-3b+6=0\\\\b=6/3=2\\\\a=6-1.5b=6-1.5*2=6-3=3$

The maximum area is when the largest side of the rectangle is 3 yards and the shortest side of the rectangle is 2 yards.