Vaishvii wrote:
JeffTargetTestPrep wrote:
houston1980 wrote:
Marie wished to enclose a rectangular region in her backyard using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides. If Marie chooses the dimensions of the rectangular region so that the region has the greatest area, what is the length, in feet, of the side of the rectangular region that is bounded by her house?
(A) 10
(B) 20
(C) 25
(D) 40
(E) 50
If the side that is bounded by the house has a length of x, then the opposite side is also x, and each of the adjacent sides has a length of (80 – x)/2.
The area of the rectangular region is:
x(80 – x)/2 = (-1/2)x^2 + 40x
The quadratic expression above takes its maximum value if:
x = -b/(2a) = -40/[2(-1/2)] = -40/(-1) = 40
Answer: DJeff, isn't square a rectangle with biggest area? Shouldnt all sides be equal?
Vaishvii you are correct that if there is a fixed total for the perimeter, then a square produces the maximum area. This thought was my first instinct on this problem too, but we have to read carefully:
houston1980 wrote:
using part of her 50-foot long house as 1 side and a total of 80 feet of fencing for the other 3 sides.
So, the fixed total of 80 feet of fencing is only for
"the other 3 sides". And the 4th side can be up to 50 feet.
Once we realize this, I find that the quickest approach is to work backwards from the answer choices to find the maximum, as
gmatophobia showed above, and I've repasted it below.
My number sense instinct said that the answer was most likely 40, since we essentially get that wall "for free", without having to use any of the 80 feet of fencing. The problem with the answer "50" is that it is too long and narrow and therefore produces a smaller area — this is the concept you were talking about.
(Optional timesaver: we don't need to test answers A and B — once we see that C is lower than D, we know that these more extreme values will be lower than C, because the graph of a quadratic equation is a "parabola." When there's a maximum, it's an upside-down U-shape, pasted below for your reference.)
Attachments
2023-11-17 16_34_20-Marie wishes to enclose a rectangular region in her backyard using _ Problem Sol.png [ 351.02 KiB | Viewed 6743 times ]
Area vs Width Parabola Graph.png [ 111.21 KiB | Viewed 6736 times ]