Hi Ben here and this is my B.O.B.

Well I'm doing fine in class, I understand the concepts and everything but I don't know if this is appropriate to ask in a Trig B.O.B but can you (Mr.K) explain the Parabola questions like the farmer with 600m of fence and he needs to make a largest possible fence question. I'm really having trouble with those questions. Ya so, that's my B.O.B. and sorry if it's too short but I'll post something on the blog or ask on the chatbox if I need extra help.

Good Luck on the test Thursday : )

## October 10, 2006

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ReplyDeleteIf I understand the question you have say 600m of fencing. There are lots of different sized rectangles you could make with that: 100x200, 50x250, etc. Each of those perimeters equals 600, and each one has a different area. Which one has the biggest area?

Now generalize so that one of the sides has a length of x. Its opposite side also has a length of x. (This is 2x meters of fence so far.) Now the critical question: how much fence is left? You had 600 and used 2x so 600-2x is left. That includes both of the other sides though so how much is just one of them? (300-x)? Yeah?

The dimensions of this generalized rectangle are x by (300-x). Multiply that out to find the area and you have a quadratic. a(x)= -x^2+300x.

Since this quadratic represents all possible rectangles with side length x, use the resulting parabola to find the maximum a(x) value. (Remember, a(x) here is the area for a given x.)

This can be done graphically or algebraically if you know how to complete the square and get your a(x) in vertex form.

Anyway you do it you will find your maximum at x = 150 and the area a(150) = 22500.

Hopefully this reminds you of your Geometry studies. Given a certain perimeter the largest area will be in the form of a square. Notice our answer of 150 is the length of one side of a square with perimeter 600.

An interesting extension that you could work out would be to start with 600 meters of fence but one side of your rectangle is already fixed (like you're building a corall along an already existing fence). The maximum area here is not given by a square.

I found that helpfull to me too.

ReplyDeletethx reversearp, and Ben for asking