Expert: Paul Klarreich - 4/9/2006

Question
My name is Janette

I am studying maths education andI know this is simple but I want to check myself.

I tried making the area equal 4 times x(3a^2 - x^2). Then I found dAdx = 3a^2 - 3x^2.
This gives a max area of 8a^3.

I'm worried that the question asks for an area and I get an expression for the area.

The question is:

Find the area of the rectangle of greatest area that has one side on the x axis and the vertices of the opposite side on the curve

y=3a^2 - x^2, -((3a)^(1/2))<x<((3a)^(1/2))

Thank-you for your help.

Janette

Answer
Hi, Janette,

You wrote:

I am studying math education and I know this is simple but I want to check myself.

I tried making the area equal 4 times x(3a^2 - x^2). Then I found dAdx = 3a^2 - 3x^2.
This gives a max area of 8a^3.

I'm worried that the question asks for an area and I get an expression for the area.

The question is:

Find the area of the rectangle of greatest area that has one side on the x axis and the vertices of the opposite side on the curve

y=3a^2 - x^2, -((3a)^(1/2))<x<((3a)^(1/2))

>> Did you write that carefully?  
 1. Is that <=, rather than < ?
 2. The intercepts would be +- a sqrt(3), not sqrt(3a)

And a more compact way to write it is: |x| <= a sqrt(3)
<<
Thank-you for your help.

Janette
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Hmmm..  Your problem asks for the largest rectangle inscribed in the parabola:

y = 3a^2 - x^2.  OK, then, you found that the area would be base times height, of course, where the base is  2x, and the height was y, which is 3a^2 - x^2.  The problem requires a diagram, but this crude method of communication prevents my sending you one. (or vice versa)

So you wrote:

A = 2x(3a^2 - x^2) = 2(3a^2 x - x^3)  and

[oops -- you seem to have 4x instead of 2x.  Where did you get an extra factor of 2?]

dA/dx = 2(3a^2 - 3x^2), and then  got  x = +- a.

Then A = 2(3a^3 - a^3) = 2(2a^3) = 4a^3.  
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Now about your 'worry'.  Not to worry.  Your problem is just poorly worded by the textbook writer.  They do it all the time.  They think that careful and precise wording is needed for their research papers on Teichmuller spaces and nonlinear Kleinian groups, but not for their textbooks.  Your problem that said:

Find the area of the rectangle of greatest area that has one side on the x axis and the vertices of the opposite side on the curve
y=3a^2 - x^2

should have said:

Find the area of the rectangle of greatest area that has one side on the x axis and the vertices of the opposite side on A MEMBER OF THE CLASS OF curves GIVEN BY the EQUATIONS:
y=3a^2 - x^2

Get the idea?  There are MANY parabolas having the FORM

y = 3a^2 - x^2

such as:

y = 3 - x^2, for a = 1
y = 12 - x^2, for a = 2
y = 300 - x^2, for a = 10

etc.

So you are solving an infinite class of problems on one sheet of paper.
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I am glad to see you are studying math education (we don't say 'maths' in this country) and wish you well in your career.  Please feel free to send me more of your 'worries'.