Question 93421This question is from textbook addison-wesley
: hello,
I would be extremely grateful for any help or advice for this problem. It has been extremely difficult for me as there is no fore-explanation on how to solve this type of problem.
it is in the section of Maximum and Minimum problems, I was not sure where to put it, I hope I got the right topic.
QUESTION:
A straight section of railroad track crosses two highways at points that are 400m and 600m, respectively, from an intersection.
Determine the Dimensions of the largest rectangular lot that can be laid out in the triangle formed by the railroad and highways.
now, I worked a few hours on this problem, coming to absolutely nothing.
I found the remaining side of the right triangle, at 721.1,
I tried any number of equations using variables, in the end coming up with 4(x-1/4L)²-1/2 meaning zilch without knowing L.
I tried graphing it and solving from there, coming up with a rough esimate of the answer.
I Know how to deal with turning into standard form and all that, but I just cannot seem to grasp which numbers I can use to come up with the (x²+1x+1)format, ... as you can see, I am very confused, any help that you can give will a lifesaver
yours truly,
Michael Demski
ps The answer from the book is 200m by 300m, I just don't know how to get there
This question is from textbook addison-wesley
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Put the intersection on the origin of an xy coordinate system.
Mark the point (600,0) on the x axis.
Mark the point (0,400) on the y axis.
Draw a line thru those two points.
That line is the railroad track.
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Find the equation of that line using the two points.
You get slope = -400/600 = -2/3
You have the y-intercept at 400
The equation is y = (-2/3)x+400
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Sketch a rectangle inside the figure touching the railroad line at (x,y),
with other vertices, (0,0), (0,y), (x,0)
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The area of that rectangle is x*y
A = xy
But y = (-2/3)x+400, so you get
A = x[(-2/3)x+400]
A = (-2/3)x^2 + 400x
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The maximum value for A occurs when x = -b/2a
x = -(400/[2(-2/3)]
x = 200/(2/3)
x = 300
Substitute that into y = (-2/3)x + 400 to get:
y = (-2/3)*300 + 400
y = -200 + 400
y = 200
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The dimensions of the rectangle with maximum area are 300 by 200
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Cheers,
Stan H.
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