Question 388328: A science museum is going to put an outdoor restaurant along one wall of the museum. The restaurant space will be rectangular. Assume the museum would prefer to maximze the area for the restaurant.
a.) Suppose there is 120 feet of fencing available for the three sides that requires fenncing. How long will the longest side of the restaurant be?
b.) What is the maximum area?
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Remember the formula for a parabola, y=(x-h)^2+k, with h & k being the coordinates of the vertex. Put the problem in this form for the area and you got it!
Let x be the long side of the rectangle area, and y the two short sides.
perimeter=120=x+2y
2y=(120-x)
y=(120-x)/2
Area=xy=x(120-x)/2=1/2(120x-x^2)=-1/2(x^2-120x)
because the coefficient is negative, this is a parabola that turns downward
completing the square, -1/2(x^2-120x+3600)+1800 (we are adding (-1800) so we must add (+1800))
=-1/2(x-60)^2+1800
referring to the basic parabola form above, h=60, and k=1800,representing the long side and maximum area respectively.
y=(120-x)/2=(120-60)/2=60/2=30
ans:(a) The long side of the restaurant is 60 feet.
(b)The maximum area of the restaurant is 1800 sq feet.
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