Craig has 140 yards of fencing with which to make a rectangular garden area. One side of the
rectangular garden will be the side of the Craig's
house. Find the length and width for a maximum
garden area.
*More than one answer may be correct; check all that apply.

BELOW ARE ALL OPTIONS:
Maximum Area: 2450 yards^2

Dimensions: 35 yards by 70 yards

Maximum Area: 1225 yards^2

Maximum Area: 4900 yards^2

Dimensions: 35 yards by 35 yards

Dimensions: 35 yards by 140 yards

9514 1404 393

Step-by-step explanation:

If the side parallel to the house is x, then the sides perpendicular to the house are each half of the remaining fence: (140-x)/2.

The area is the product of length (x) and width (140-x)/2, so is ...

  A = x(140-x)/2

This is the equation of a parabola that opens downward. Its vertex (maximum) will lie where x is halfway between the zeros of 0 and 140. That is, x=70 will give the largest area. The width of the garden area at that point will be ...

  (140 -70)/2 = 35

So, the dimensions of the garden with maximum area are 35 yards by 70 yards. The maximum area is the product of these dimensions, 2450 yd².

_____

Additional comment

The maximum area will be found where the cost of fence parallel to the house is equal to the cost of fence perpendicular to the house. Here, that's 70 yards of fence (half the 140 yd total) in each of those directions. The same remains true if some of the fence is used for partitioning the interior of the fenced space. (Of course, if fence is used on all sides, the shape becomes a square--unless cost factors cause it to be otherwise.)

We find this problem a bit interesting in that a house 210 feet long is somewhat unusual.