SOLUTION: A rectangular garden is enclosed so that the river borders 1 side and 300m of fencing are used for the other 3 sides. a)What is the formula for the area in terms of x and y? b)

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Question 286594: A rectangular garden is enclosed so that the river borders 1 side and 300m of fencing are used for the other 3 sides.
a)What is the formula for the area in terms of x and y?
b)What is the formul that relates x and y to the perimeter?
c)Write a formulaa for the area in terms of x only
d)What are the dimensions of the garden if the area is a maximum?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let the length of the side bordering the river = y
Let the 2 sides perpendicular to the side bordering the river = x
given:
a)A+=+xy m2
P+=+300+%2B+y
b)P=+2x+%2B+2y m
----------------
2x+=+300+-+y
y+=+300+-+2x
A+=+x%2A%28300+-+2x%29
c)A+=+300x+-+2x%5E2
x is a maximum halfway between the 2 roots. First I'll find the roots
x%2A%282x+-+300%29+=+0
The roots are x+=+0 and
2x+-+300+=+0
2x+=+300
x+=+150
The midpoint is x+=+75
y+=+300+-+2x
y+=+300+-+150
y+=+150
d)The dimensions that maximize area are 75 x 150
check:
Just vary the dimensions a little bit, and Area should decrease
A+=+75%2A150
A+=+11250 m2
Suppose:
A2+=+76%2A148 -note that 2%2A76+%2B+148+=+152+%2B+148
and 152%2B+148+=+300 as it should
A2+=+11248 It decreased by 2 m
OK