John G. answered • 10/25/15
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Understanding math via the real world.
When I draw a picture of this rectangular fence, I think of the width as coming off the river and the length running parallel to it, so I have a box with a river side, two x's and one y.
The total fence I use is 296, but it's also 2x + y, so 2x + y = 296.
The total area is just A = x*y since it's a rectangle and we can use length times width.
The problem is we can't really use the area equation because it has too many variables, so we need to do some substitution. If we solve the first equation for y by subtracting 2x on each side, we should get y = 296 - 2x. Then we can plug that in for y in the area equation and get A = x*(296 - 2x), you can distribute and rearrange if you like and get a quadratic equation that looks like A = -2x2 + 296x which will give us a nice upside down parabola when graphed, meaning it has a maximum value.
To find that maximum we can use the equation for the axis of symmetry x = -b/(2a), and plug in a = -2 and b = 296. This will find our x value where the rectangle has the biggest area. We can then plug that into y = 296 - 2x to find the y value. Multiply them to find the area.
Erin M.
And also Y= P-2X I forgot to put that with the problem
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10/25/15
John G.
The question seems to be looking for the area. "find the dimensions (width x and length y) of the plot that will maximize the area."
You're given the perimeter of 296.
You're looking for the dimensions (x and y) and then also the largest area, no?
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10/25/15
Erin M.
Yes I'm sorry you are right but I did forget to mention the Y= P-2x
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10/25/15
John G.
That's ok. I used it, just took a minute to get to because I didn't know you were given that. It's where I say that y = 296 - 2x. Then you plug that into the area equation to make your parabola.
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10/25/15
Erin M.
Thank you! That was right and was very helpful!
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10/25/15
Erin M.
10/25/15