9514 1404 393
Answer:
- Maximum Area: 2450 yards^2
- Dimensions: 35 yards by 70 yards
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.
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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.